[{"data":1,"prerenderedAt":93115},["ShallowReactive",2],{"topic-finance":3},[4,145,463,1021,41657,67674,72283,72426,73114,73761,73806,74140,74479,75071,75400,76262,76944,77818,78423,79101,79497,80176,80947,81075,81723,82617,83498,84153,84606,85201,85756,86018,86359,87148,88421,88457,90115,91062,91641,91676,91786,92214,92329,92816],{"id":5,"title":6,"authors":7,"body":9,"breadcrumb":111,"builders":115,"byline":116,"challenge":117,"courseAuthor":116,"courseLead":116,"dek":123,"description":124,"draft":125,"extension":126,"eyebrow":127,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":128,"heroImage":116,"kind":131,"lessonCount":116,"meta":132,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":134,"publishDate":135,"readingTime":116,"related":136,"relatedProjects":116,"seo":137,"stem":140,"tags":141,"track":116,"trackName":116,"__hash__":144},"blog\u002Fblog\u002Fprograms\u002Fcredits-grant.md","Qollab Grant Program",[8],"nico",{"type":10,"value":11,"toc":103},"minimark",[12,17,21,24,28,37,41,75,82,86,93],[13,14,16],"h2",{"id":15},"get-rewarded-for-helping-quantum-grow","Get rewarded for helping quantum grow",[18,19,20],"p",{},"Whether collecting dust in GitHub repos hardly anyone visits, or published once in a paper and never touched again, the quantum ecosystem's best work is often scattered and hard to find. Qollab exists to change that: we're building the one place where the open-source community can learn about quantum as well as publish, fork, and run quantum code on real hardware.",[18,22,23],{},"To make progress on our mission, we need your help — and we are willing to reward you for it. Port your already published circuits and tutorials onto Qollab, fork and extend some of our existing open-source projects, or bring over new original work, and you'll be eligible for quantum compute credits you can use on any project of your choosing.",[13,25,27],{"id":26},"how-you-can-earn-credits","How you can earn credits",[29,30],"benefit-grid",{"b1":31,"b2":32,"b3":33,"t1":34,"t2":35,"t3":36},"You've already developed notebooks, tutorials, algorithms and circuits. Clean them up, have them run natively on Qollab, and you could earn up to $10,000 in compute credits per project. Because the work already exists, review cycles are fast — free access to real quantum hardware in record time.","Take an existing open-source Qollab project and build on it: add new features, new circuits, or take it in a meaningfully different direction. If you are unsure about your approach, the original developers are usually a message away.","Have an itch you want to scratch? Build new-to-the-world tutorials, notebooks and circuit implementations, and earn up to $20,000 in compute credits for content that helps the community learn and grow.","Port existing work — up to $10,000","Fork and extend — up to $10,000","Publish original work — up to $20,000",[13,38,40],{"id":39},"how-the-process-works","How the process works",[42,43,44,54,57,60,63,66,69,72],"ul",{},[45,46,47,48,53],"li",{},"You submit your proposal using ",[49,50,52],"a",{"href":51},"https:\u002F\u002Fairtable.com\u002FappMaDEaU08NjHUfu\u002Fpag7Fof6hQGzNpDtu\u002Fform","this form",".",[45,55,56],{},"Our team reviews every submission on a rolling basis.",[45,58,59],{},"We'll tell you where your submission lands before you commit to the work.",[45,61,62],{},"Projects need to be published on Qollab and run natively on the platform.",[45,64,65],{},"All work needs to be under the MIT license, so everyone can fork and build on it.",[45,67,68],{},"Once your project is live and runnable, credits land on your Qollab account.",[45,70,71],{},"Credits expire within 3 months, to make sure they get used rather than banked.",[45,73,74],{},"Your work needs to stay on Qollab for at least a year from the day you published it.",[18,76,77,78,53],{},"Full terms, including eligibility and how credits are issued and expire, are in the ",[49,79,81],{"href":80},"\u002Fprograms\u002Fcredits-terms","Grant Program Terms & Conditions",[13,83,85],{"id":84},"questions","Questions",[18,87,88,89,53],{},"Reach out at ",[49,90,92],{"href":91},"mailto:hello@qollab.xyz","hello@qollab.xyz",[94,95,100],"cta-band",{"f1":96,"f2":80,"l1":97,"l2":98,"title":99},"https:\u002F\u002Fairtable.com\u002FappMaDEaU08NjHUfu\u002FpagfidAqW1fppbUs4\u002Fform","Custom program interest form","Read the terms","Have a bigger idea?",[18,101,102],{},"Running a student club, a research lab, an open-source project, or any other community that wants real quantum hardware access for its members? We build custom versions of this programme for groups, not just individuals. Tell us about your community and what you have in mind.",{"title":104,"searchDepth":105,"depth":105,"links":106},"",2,[107,108,109,110],{"id":15,"depth":105,"text":16},{"id":26,"depth":105,"text":27},{"id":39,"depth":105,"text":40},{"id":84,"depth":105,"text":85},[112,113,114],"Qollab","Programs","Grant Program",[],null,{"type":118,"status":119,"deadline":120,"prize":121,"terms":122},"funding","open","Rolling — every submission reviewed as it arrives","Up to $20,000 in IonQ compute credits per project","MIT licensed, published and runnable on Qollab for one year","The quantum ecosystem's best work is scattered across repos nobody visits and papers nobody re-runs. Bring it to Qollab — port it, fork it, or build something new — and earn compute credits you can spend on real quantum hardware.","Port, fork, or publish open-source quantum work on Qollab and earn up to $20,000 in IonQ compute credits. Submissions reviewed on a rolling basis.",false,"md","Open · rolling review",{"primaryHref":51,"primaryLabel":129,"secondaryHref":80,"secondaryLabel":98,"note":130},"Submit a proposal →","Three ways to qualify, reviewed on a rolling basis. We tell you where your project lands before you commit to the work.","challenge",{},true,"\u002Fblog\u002Fprograms\u002Fcredits-grant","2026-07-30",[],{"title":138,"description":139},"Qollab Grant Program: earn quantum compute credits","Earn up to $20,000 in IonQ compute credits for porting, forking, or publishing open-source quantum work on Qollab. MIT licensed, runnable on real hardware.","blog\u002Fprograms\u002Fcredits-grant",[118,142,143],"ionq","quantum","P7fj7e3AOcEpPSzUbFsCKAtMfoh2zUAFOOxmM9LpkOM",{"id":146,"title":147,"authors":148,"body":149,"breadcrumb":449,"builders":451,"byline":116,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":452,"description":453,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":116,"lessonCount":116,"meta":454,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":455,"publishDate":135,"readingTime":116,"related":456,"relatedProjects":116,"seo":457,"stem":460,"tags":461,"track":116,"trackName":116,"__hash__":462},"blog\u002Fblog\u002Fprograms\u002Fcredits-terms.md","Grant Program: Terms & Conditions",[8],{"type":10,"value":150,"toc":436},[151,158,161,165,190,194,236,240,254,258,283,287,304,308,336,340,351,355,366,370,378,382,395,399,431],[18,152,153,157],{},[154,155,156],"strong",{},"Last updated:"," July 30, 2026",[18,159,160],{},"These Terms & Conditions (\"Terms\") govern participation in the Qollab Grant Program (the \"Program\"), operated by fluxDev, Inc. By submitting work to the Program, you (\"Contributor,\" \"you\") agree to these Terms.",[13,162,164],{"id":163},"_1-eligibility","1. Eligibility",[42,166,167,170,184,187],{},[45,168,169],{},"You must be at least 18 years old, or the age of legal majority in your jurisdiction, to participate.",[45,171,172,173],{},"The Program is not open to:\n",[42,174,175,178,181],{},[45,176,177],{},"Current employees, contractors, or advisors of Qollab",[45,179,180],{},"Immediate family members of the above",[45,182,183],{},"Residents of, or entities based in, jurisdictions subject to applicable export control or trade sanctions restrictions (see Section 9)",[45,185,186],{},"Qollab reserves the right to verify eligibility at any point before or after credits are awarded, and to disqualify a Contributor who does not meet these requirements, including retroactively.",[45,188,189],{},"There is no limit on the number of submissions a single Contributor may make.",[13,191,193],{"id":192},"_2-submission-requirements","2. Submission requirements",[42,195,196,216,233],{},[45,197,198,199],{},"To be eligible for credits, submitted work must:\n",[42,200,201,204,207,210,213],{},[45,202,203],{},"Be published natively and run on the Qollab platform (qollab.xyz)",[45,205,206],{},"Be fully runnable: one-click executable on the platform, not a static code listing",[45,208,209],{},"Be licensed under the MIT License at the time of publication",[45,211,212],{},"Fall into one of the three recognised categories on the Program page: Port Existing Work, Fork & Extend, or Publish Original Work",[45,214,215],{},"Remain published and runnable natively on Qollab for a minimum of one (1) year from the date of publication, except where removal is required as set out in Section 3, or as requested by Qollab",[45,217,218,219],{},"By submitting work, you represent and warrant that:\n",[42,220,221,224,227,230],{},[45,222,223],{},"You are the original author of the submitted work, or you hold all rights necessary to publish it under an MIT license and to grant the rights described in Section 3",[45,225,226],{},"The work does not infringe any third party's copyright, patent, trade secret, or other intellectual property rights",[45,228,229],{},"The work does not violate any applicable law or the rights of any third party",[45,231,232],{},"Where the work is a \"port\" of pre-existing material (for example, from a published paper, prior notebook, or other repository), you have the right to republish that material and, where the original author is someone other than you, you have secured their consent or the material is otherwise appropriately licensed for this use",[45,234,235],{},"Qollab may request evidence of authorship or rights of use before or after publication, and may remove any submission and revoke any associated award if such evidence is not provided or is found to be false.",[13,237,239],{"id":238},"_3-intellectual-property-and-license-grant","3. Intellectual property and license grant",[42,241,242,245,248,251],{},[45,243,244],{},"You retain ownership of the copyright in your submitted work. Nothing in these Terms transfers ownership of your work to Qollab.",[45,246,247],{},"By publishing work through the Program, you grant Qollab a non-exclusive, worldwide, royalty-free, sublicensable license to host, display, promote, format, and technically adapt (for example, for platform compatibility) your published work on and in connection with Qollab, for as long as the work remains published on the platform — minimum one year from the date of publication. Where a Contributor is required to remove published work prior to the one-year minimum due to a valid third-party legal claim (for example, a substantiated infringement claim) or a court order, such removal will not be treated as a breach of this Section, provided the Contributor promptly notifies Qollab of the removal and its cause.",[45,249,250],{},"The MIT license under which your work is published governs third-party use, including forking and extension by other developers. You cannot revoke or narrow this license retroactively for versions already forked or distributed by others.",[45,252,253],{},"You may request removal of your work from Qollab at any time after a calendar year. Removal does not affect the rights already granted to third parties under the MIT license for versions forked prior to removal, and Qollab reserves the right to claw back any associated unused credits per Section 5.",[13,255,257],{"id":256},"_4-review-and-award-process","4. Review and award process",[42,259,260,263,277,280],{},[45,261,262],{},"All submissions are reviewed by Qollab on a rolling basis.",[45,264,265,266],{},"Qollab has sole discretion to:\n",[42,267,268,271,274],{},[45,269,270],{},"Accept, reject, or request revisions to any submission",[45,272,273],{},"Determine which category a submission qualifies for",[45,275,276],{},"Determine the credit amount awarded, up to the maximum published for that category on the Program page",[45,278,279],{},"Review decisions are final. There is no appeals process for the amount of credit awarded or for a decision not to accept a submission. This does not affect any separate disqualification process described in Section 6.",[45,281,282],{},"Qollab is not obligated to provide detailed reasoning for a rejection, though we will make reasonable efforts to give general feedback where useful to the Contributor.",[13,284,286],{"id":285},"_5-credits-issuance-use-and-expiration","5. Credits: issuance, use, and expiration",[42,288,289,292,295,298,301],{},[45,290,291],{},"\"Compute credits\" refers to allocated usage credit redeemable for quantum hardware time on IonQ systems accessible through Qollab. Credits are denominated in USD value and applied toward compute usage; they are not cash and have no cash redemption value.",[45,293,294],{},"Credits are awarded to the Contributor's Qollab account upon publication approval and are non-transferable. They cannot be sold, gifted, pooled with another Contributor's credits, or redeemed for cash.",[45,296,297],{},"Credits expire three (3) months from the date of issuance. Unused credits are forfeited automatically upon expiration and cannot be reinstated, extended, or converted to any other benefit, except at Qollab's sole discretion.",[45,299,300],{},"Qollab does not guarantee the continuous availability of compute capacity within the 3-month usage window. In the event of a hardware or platform outage that materially prevents a Contributor from using awarded credits before expiration, Qollab will, at its discretion, extend the expiration window or reissue equivalent credits.",[45,302,303],{},"Credit amounts, categories, and award maximums described on the Program page are subject to change for future submissions at Qollab's discretion. Changes do not affect credits already awarded.",[13,305,307],{"id":306},"_6-disqualification-and-clawback","6. Disqualification and clawback",[42,309,310,330,333],{},[45,311,312,313],{},"Qollab may disqualify a submission and revoke associated credits, whether before or after issuance, if it determines that:\n",[42,314,315,318,321,324,327],{},[45,316,317],{},"The submission does not meet the requirements of Section 2",[45,319,320],{},"The Contributor made false representations under Section 2",[45,322,323],{},"The work is later found to infringe third-party rights",[45,325,326],{},"The Contributor violated these Terms in any other material way",[45,328,329],{},"The Contributor removes, unpublishes, or materially disables the submitted work from Qollab before the one-year minimum publication period in Section 2 has elapsed, other than as permitted under Section 3, or at Qollab's own request",[45,331,332],{},"If credits have already been used at the time of a confirmed disqualification, Qollab reserves the right to invoice the Contributor for the value of credits used, or to take other reasonable steps to recover that value, at Qollab's discretion.",[45,334,335],{},"Disqualification determinations under this Section are final and not subject to appeal regarding the underlying award amount, though a Contributor may request reconsideration solely on the question of whether a disqualifying condition in this Section was met.",[13,337,339],{"id":338},"_7-taxes","7. Taxes",[42,341,342,345,348],{},[45,343,344],{},"Credits awarded under the Program may constitute taxable income to the Contributor under applicable law. Contributors are solely responsible for determining and meeting any tax obligations arising from participation in the Program.",[45,346,347],{},"Where required by applicable law (including for US-based Contributors receiving cumulative awards above applicable reporting thresholds), Qollab may request tax information (for example, a completed Form W-9 or W-8BEN) prior to issuing credits, and may issue applicable tax forms (for example, Form 1099) reflecting the value of credits awarded.",[45,349,350],{},"Failure to provide requested tax information may result in delayed or withheld issuance of credits.",[13,352,354],{"id":353},"_8-program-changes-and-termination","8. Program changes and termination",[42,356,357,360,363],{},[45,358,359],{},"Qollab may modify, suspend, or terminate the Program, in whole or in part, at any time and without prior notice. This includes changes to eligible categories, credit amounts, review criteria, or expiration terms for future submissions.",[45,361,362],{},"If the Program is terminated, credits already issued and not yet expired remain valid and usable per Section 5, subject to compute availability under Section 5.",[45,364,365],{},"Submissions in the review queue at the time of Program suspension or termination will be handled at Qollab's discretion, which may include continued review, deferral, or closure without award.",[13,367,369],{"id":368},"_9-compliance-and-export-controls","9. Compliance and export controls",[42,371,372,375],{},[45,373,374],{},"Access to quantum hardware through the Program may be subject to U.S. export control laws and regulations, and comparable laws in other jurisdictions. Contributors are responsible for complying with all applicable export control and sanctions laws.",[45,376,377],{},"Qollab may deny or revoke participation for any Contributor located in, or affiliated with, a jurisdiction or entity subject to applicable trade restrictions.",[13,379,381],{"id":380},"_10-privacy","10. Privacy",[42,383,384,392],{},[45,385,386,387,391],{},"Information collected through the Program submission form (including name, contact information, and submission details) is handled in accordance with ",[49,388,390],{"href":389},"\u002Fprivacy","Qollab's Privacy Policy",". The Program's submission forms are hosted on Airtable, so information you enter into them is collected through that service.",[45,393,394],{},"Published Contributor names and public profile information associated with submitted work will be visible to other users of the platform as part of normal platform functionality.",[13,396,398],{"id":397},"_11-general","11. General",[42,400,401,407,413,419,425],{},[45,402,403,406],{},[154,404,405],{},"No guarantee of future participation."," Acceptance of a past submission does not guarantee acceptance of future submissions.",[45,408,409,412],{},[154,410,411],{},"Force majeure."," Qollab is not liable for any failure or delay in Program operation, credit issuance, or compute availability caused by circumstances beyond its reasonable control, including platform outages, hardware partner disruptions, or acts of God.",[45,414,415,418],{},[154,416,417],{},"No warranty."," The Program and any associated compute access are provided \"as is,\" without warranty of any kind, to the extent permitted by applicable law.",[45,420,421,424],{},[154,422,423],{},"Governing law."," These Terms are governed by the laws of California, without regard to conflict of law principles.",[45,426,427,430],{},[154,428,429],{},"Changes to these Terms."," Qollab may update these Terms from time to time. Material changes will be reflected by an updated \"Last updated\" date above. Continued participation in the Program after changes take effect constitutes acceptance of the revised Terms.",[18,432,433,434,53],{},"Questions about these Terms can be directed to ",[49,435,92],{"href":91},{"title":104,"searchDepth":105,"depth":105,"links":437},[438,439,440,441,442,443,444,445,446,447,448],{"id":163,"depth":105,"text":164},{"id":192,"depth":105,"text":193},{"id":238,"depth":105,"text":239},{"id":256,"depth":105,"text":257},{"id":285,"depth":105,"text":286},{"id":306,"depth":105,"text":307},{"id":338,"depth":105,"text":339},{"id":353,"depth":105,"text":354},{"id":368,"depth":105,"text":369},{"id":380,"depth":105,"text":381},{"id":397,"depth":105,"text":398},[112,113,450],"Grant Program Terms",[],"The terms governing participation in the Qollab Grant Program.","Terms and Conditions for the Qollab Grant Program — eligibility, submission requirements, licensing, review, and how compute credits are issued, used and expire.",{},"\u002Fblog\u002Fprograms\u002Fcredits-terms",[],{"title":458,"description":459},"Qollab Grant Program: Terms & Conditions","Eligibility, submission requirements, IP and licensing, review and award process, and credit issuance and expiry for the Qollab Grant Program.","blog\u002Fprograms\u002Fcredits-terms",[],"CYOJeXS_GFTxWN7MZSvxcjGEdtiUfOQnvEHVqZ6PrnI",{"id":464,"title":465,"authors":466,"body":468,"breadcrumb":968,"builders":972,"byline":983,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":985,"description":986,"draft":125,"extension":126,"eyebrow":987,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":988,"lessonCount":116,"meta":989,"navigation":133,"newsItems":116,"next":116,"ogImage":990,"order":116,"outcomes":116,"path":991,"publishDate":992,"readingTime":993,"related":994,"relatedProjects":995,"seo":1015,"stem":1017,"tags":1018,"track":116,"trackName":116,"__hash__":1020},"blog\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization.md","2-qubit states, visualized 8 ways",[467],"onri-jay-benally",{"type":10,"value":469,"toc":956},[470,473,485,489,492,820,823,827,830,837,840,846,850,853,859,863,866,872,876,879,885,889,892,896,899,905,909,912,918,922,925,931,937,941,952],[18,471,472],{},"A 2-qubit state is a four-dimensional complex vector. That's easy to write down and hard to picture, and entanglement makes it harder: the most interesting thing about the state is precisely the part you can't see by looking at each qubit alone. This walkthrough takes one small circuit and renders its state eight different ways, because each visualization makes a different property visible.",[18,474,475,476,480,481,53],{},"Everything below comes from Onri's runnable notebook, shared openly on ",[49,477,479],{"href":478},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","GitHub",". The code and figures are his, republished here with permission. You can run the whole thing end to end in ",[49,482,484],{"href":483},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","Google Colab",[13,486,488],{"id":487},"one-circuit-two-states","One circuit, two states",[18,490,491],{},"The experiment is a controlled comparison. First, the baseline: the product state |00⟩, where each qubit has its own well-defined state and nothing is correlated. Then the same eight visualizations run again on an entangled state, built with a Hadamard, a CNOT, and a couple of single-qubit rotations:",[493,494,497],"code-block",{"name":495,"tag":496},"2_qubit_state_visualization.ipynb","Python",[498,499,503],"pre",{"className":500,"code":501,"language":502,"meta":104,"style":104},"language-python shiki shiki-themes one-dark-pro","import numpy as np\nfrom qiskit import QuantumCircuit\nfrom qiskit.quantum_info import Statevector, DensityMatrix\n\n# Create a circuit with 2 qubits\nqc = QuantumCircuit(2)\nqc.h(0)             # Hadamard on q0\nqc.cx(0, 1)         # CNOT with q0 as control, q1 as target\nqc.ry(np.pi \u002F 4, 1) # Y-rotation of pi\u002F4 on q1\nqc.z(1)             # Phase flip (Z gate) on q1\n\nfinal_statevector = Statevector(qc)\nfinal_density_matrix = DensityMatrix(qc)\n\n# Generate all 8 plots for the final state\nplot_bloch_multivector_viz(final_density_matrix, \"Final Entangled State\")\nplot_q_sphere_viz(final_density_matrix, \"Final Entangled State\")\nplot_steering_ellipsoid(final_density_matrix, \"Steering Ellipsoid for Final State\")\nvisualize_schmidt_decomposition(final_statevector, \"Schmidt Decomposition for Final State\")\nexplain_toroidal_geometry()\nplot_correlation_invariants(final_density_matrix, \"Final Entangled State\")\nplot_wigner_function(final_density_matrix, \"Wigner Functions for Final State\")\nplot_density_matrix(final_density_matrix, \"Density Matrix for Final State\")\n","python",[504,505,506,525,538,551,557,564,588,608,632,659,676,681,695,708,713,719,734,746,759,773,782,794,807],"code",{"__ignoreMap":104},[507,508,511,515,519,522],"span",{"class":509,"line":510},"line",1,[507,512,514],{"class":513},"seHd6","import",[507,516,518],{"class":517},"sn6KH"," numpy ",[507,520,521],{"class":513},"as",[507,523,524],{"class":517}," np\n",[507,526,527,530,533,535],{"class":509,"line":105},[507,528,529],{"class":513},"from",[507,531,532],{"class":517}," qiskit ",[507,534,514],{"class":513},[507,536,537],{"class":517}," QuantumCircuit\n",[507,539,541,543,546,548],{"class":509,"line":540},3,[507,542,529],{"class":513},[507,544,545],{"class":517}," qiskit.quantum_info ",[507,547,514],{"class":513},[507,549,550],{"class":517}," Statevector, DensityMatrix\n",[507,552,554],{"class":509,"line":553},4,[507,555,556],{"emptyLinePlaceholder":133},"\n",[507,558,560],{"class":509,"line":559},5,[507,561,563],{"class":562},"sV9Aq","# Create a circuit with 2 qubits\n",[507,565,567,570,574,578,581,585],{"class":509,"line":566},6,[507,568,569],{"class":517},"qc ",[507,571,573],{"class":572},"sjrmR","=",[507,575,577],{"class":576},"sVbv2"," QuantumCircuit",[507,579,580],{"class":517},"(",[507,582,584],{"class":583},"sVC51","2",[507,586,587],{"class":517},")\n",[507,589,591,594,597,599,602,605],{"class":509,"line":590},7,[507,592,593],{"class":517},"qc.",[507,595,596],{"class":576},"h",[507,598,580],{"class":517},[507,600,601],{"class":583},"0",[507,603,604],{"class":517},")             ",[507,606,607],{"class":562},"# Hadamard on q0\n",[507,609,611,613,616,618,620,623,626,629],{"class":509,"line":610},8,[507,612,593],{"class":517},[507,614,615],{"class":576},"cx",[507,617,580],{"class":517},[507,619,601],{"class":583},[507,621,622],{"class":517},", ",[507,624,625],{"class":583},"1",[507,627,628],{"class":517},")         ",[507,630,631],{"class":562},"# CNOT with q0 as control, q1 as target\n",[507,633,635,637,640,643,646,649,651,653,656],{"class":509,"line":634},9,[507,636,593],{"class":517},[507,638,639],{"class":576},"ry",[507,641,642],{"class":517},"(np.pi ",[507,644,645],{"class":572},"\u002F",[507,647,648],{"class":583}," 4",[507,650,622],{"class":517},[507,652,625],{"class":583},[507,654,655],{"class":517},") ",[507,657,658],{"class":562},"# Y-rotation of pi\u002F4 on q1\n",[507,660,662,664,667,669,671,673],{"class":509,"line":661},10,[507,663,593],{"class":517},[507,665,666],{"class":576},"z",[507,668,580],{"class":517},[507,670,625],{"class":583},[507,672,604],{"class":517},[507,674,675],{"class":562},"# Phase flip (Z gate) on q1\n",[507,677,679],{"class":509,"line":678},11,[507,680,556],{"emptyLinePlaceholder":133},[507,682,684,687,689,692],{"class":509,"line":683},12,[507,685,686],{"class":517},"final_statevector ",[507,688,573],{"class":572},[507,690,691],{"class":576}," Statevector",[507,693,694],{"class":517},"(qc)\n",[507,696,698,701,703,706],{"class":509,"line":697},13,[507,699,700],{"class":517},"final_density_matrix ",[507,702,573],{"class":572},[507,704,705],{"class":576}," DensityMatrix",[507,707,694],{"class":517},[507,709,711],{"class":509,"line":710},14,[507,712,556],{"emptyLinePlaceholder":133},[507,714,716],{"class":509,"line":715},15,[507,717,718],{"class":562},"# Generate all 8 plots for the final state\n",[507,720,722,725,728,732],{"class":509,"line":721},16,[507,723,724],{"class":576},"plot_bloch_multivector_viz",[507,726,727],{"class":517},"(final_density_matrix, ",[507,729,731],{"class":730},"subq3","\"Final Entangled State\"",[507,733,587],{"class":517},[507,735,737,740,742,744],{"class":509,"line":736},17,[507,738,739],{"class":576},"plot_q_sphere_viz",[507,741,727],{"class":517},[507,743,731],{"class":730},[507,745,587],{"class":517},[507,747,749,752,754,757],{"class":509,"line":748},18,[507,750,751],{"class":576},"plot_steering_ellipsoid",[507,753,727],{"class":517},[507,755,756],{"class":730},"\"Steering Ellipsoid for Final State\"",[507,758,587],{"class":517},[507,760,762,765,768,771],{"class":509,"line":761},19,[507,763,764],{"class":576},"visualize_schmidt_decomposition",[507,766,767],{"class":517},"(final_statevector, ",[507,769,770],{"class":730},"\"Schmidt Decomposition for Final State\"",[507,772,587],{"class":517},[507,774,776,779],{"class":509,"line":775},20,[507,777,778],{"class":576},"explain_toroidal_geometry",[507,780,781],{"class":517},"()\n",[507,783,785,788,790,792],{"class":509,"line":784},21,[507,786,787],{"class":576},"plot_correlation_invariants",[507,789,727],{"class":517},[507,791,731],{"class":730},[507,793,587],{"class":517},[507,795,797,800,802,805],{"class":509,"line":796},22,[507,798,799],{"class":576},"plot_wigner_function",[507,801,727],{"class":517},[507,803,804],{"class":730},"\"Wigner Functions for Final State\"",[507,806,587],{"class":517},[507,808,810,813,815,818],{"class":509,"line":809},23,[507,811,812],{"class":576},"plot_density_matrix",[507,814,727],{"class":517},[507,816,817],{"class":730},"\"Density Matrix for Final State\"",[507,819,587],{"class":517},[18,821,822],{},"Every method below shows the entangled state; where the contrast with |00⟩ is the whole point, both are shown.",[13,824,826],{"id":825},"_1-two-bloch-spheres-one-giveaway","1. Two Bloch spheres, one giveaway",[18,828,829],{},"The Bloch multivector shows the state of each qubit on its own Bloch sphere. For |00⟩, both vectors reach the surface: each qubit is in a pure, definite state.",[831,832],"blog-figure",{"alt":833,"caption":834,"no":835,"src":836},"Two Bloch spheres for the initial state |00>, each with a vector pointing to the north pole","The product state |00⟩: each qubit's vector reaches the surface of its own sphere.","Fig. 1","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-initial.webp",[18,838,839],{},"For the entangled state, the vectors retreat inside the sphere. Each qubit on its own looks like a mixed state. The information isn't in either qubit, it's in the correlation between them. The shrunken vector is the visual signature of entanglement.",[831,841],{"alt":842,"caption":843,"no":844,"src":845},"Two Bloch spheres for the entangled state, each vector noticeably inside the sphere","The entangled state: both vectors pull inside the sphere; each qubit alone is mixed.","Fig. 2","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.webp",[13,847,849],{"id":848},"_2-the-q-sphere","2. The Q-sphere",[18,851,852],{},"The Q-sphere maps state amplitudes and phases onto a single sphere: the size of each point is the magnitude of the amplitude for a basis state, and its color is the phase. Where the Bloch view loses the correlations, the Q-sphere shows the full 2-qubit state at once.",[831,854],{"alt":855,"caption":856,"no":857,"src":858},"Q-sphere of the entangled state showing amplitude points with phase coloring","The entangled state on the Q-sphere: amplitude as point size, phase as color.","Fig. 3","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fqsphere-entangled.webp",[13,860,862],{"id":861},"_3-the-steering-ellipsoid","3. The steering ellipsoid",[18,864,865],{},"The steering ellipsoid shows the set of states qubit B can be \"steered\" into by measuring qubit A. If it collapses to a single point, there is no steering at all; a non-trivial ellipsoid means the state is entangled. For |00⟩ the notebook can't even draw one: the ellipsoid degenerates to a point, which is its own kind of answer.",[831,867],{"alt":868,"caption":869,"no":870,"src":871},"Quantum steering ellipsoid for the entangled state","The entangled state's steering ellipsoid: measuring qubit A can steer qubit B anywhere on this surface.","Fig. 4","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fsteering-ellipsoid-entangled.webp",[13,873,875],{"id":874},"_4-schmidt-decomposition","4. Schmidt decomposition",[18,877,878],{},"Any pure 2-qubit state can be written as |ψ⟩ = λ₀|u₀⟩|v₀⟩ + λ₁|u₁⟩|v₁⟩. The Schmidt coefficients λ₀ and λ₁ quantify entanglement directly: a product state has one coefficient equal to 1 and the other 0, while an entangled state splits the weight between both terms.",[831,880],{"alt":881,"caption":882,"no":883,"src":884},"Bar chart of the Schmidt coefficients for the entangled state","Both Schmidt coefficients are non-zero: the weight is shared, so the state is entangled.","Fig. 5","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fschmidt-entangled.webp",[13,886,888],{"id":887},"_5-toroidal-geometry","5. Toroidal geometry",[18,890,891],{},"Not everything has a standard plot. The toroidal view is a highly abstract method that maps entanglement invariants onto the surface of a torus. Powerful for classifying entanglement, but beyond standard plotting libraries. The notebook explains the concept and moves on; it's included because knowing a representation exists is half of finding a use for it.",[13,893,895],{"id":894},"_6-correlation-invariants","6. Correlation invariants",[18,897,898],{},"This plot shows the invariants of the state under local unitary operations: the eigenvalues of TᵀT, where T is the correlation matrix. These numbers don't change when either qubit is rotated on its own, so they capture the nature and strength of the correlation itself: the part of the state that no local operation can create or destroy.",[831,900],{"alt":901,"caption":902,"no":903,"src":904},"Bar plot of local correlation invariants for the entangled state","Local invariants of the entangled state: what remains when you ignore everything each qubit does alone.","Fig. 6","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fcorrelation-invariants-entangled.webp",[13,906,908],{"id":907},"_7-the-wigner-function","7. The Wigner function",[18,910,911],{},"A phase-space representation of each qubit's reduced state. Negative values, shown in red, are a key signature of non-classicality. No classical probability distribution can go below zero.",[831,913],{"alt":914,"caption":915,"no":916,"src":917},"Wigner function plots for the entangled state with negative regions shown in red","Wigner functions for the entangled state: the red, negative regions have no classical counterpart.","Fig. 7","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fwigner-entangled.webp",[13,919,921],{"id":920},"_8-the-density-matrix","8. The density matrix",[18,923,924],{},"The workhorse. Real parts in blue, imaginary parts in red; the diagonal holds the populations and the off-diagonal elements indicate coherence and entanglement. For |00⟩, a single bar stands alone.",[831,926],{"alt":927,"caption":928,"no":929,"src":930},"Density matrix of the initial state |00>, a single bar at the 00 position","The density matrix of |00⟩: one population, no coherences.","Fig. 8","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fdensity-initial.webp",[831,932],{"alt":933,"caption":934,"no":935,"src":936},"Density matrix of the entangled state with multiple bars including off-diagonal elements","The entangled state: off-diagonal structure appears, the coherences that carry the correlations.","Fig. 9","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fdensity-entangled.webp",[13,938,940],{"id":939},"run-it-yourself","Run it yourself",[18,942,943,944,947,948,951],{},"Every figure above comes out of one script you can run top to bottom: ",[49,945,946],{"href":478},"open the notebook on GitHub"," or ",[49,949,950],{"href":483},"run it directly in Colab",". Change the circuit in Step 5, run again, and watch all eight views shift together. That's where the intuition comes from.",[953,954,955],"style",{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":957},[958,959,960,961,962,963,964,965,966,967],{"id":487,"depth":105,"text":488},{"id":825,"depth":105,"text":826},{"id":848,"depth":105,"text":849},{"id":861,"depth":105,"text":862},{"id":874,"depth":105,"text":875},{"id":887,"depth":105,"text":888},{"id":894,"depth":105,"text":895},{"id":907,"depth":105,"text":908},{"id":920,"depth":105,"text":921},{"id":939,"depth":105,"text":940},[112,969,970,971],"Blog","Expert notes","2-qubit states",[973],{"username":467,"name":974,"role":975,"bio":976,"links":977},"Onri Jay Benally","Quantum Hardware Engineer, IBM · University of Minnesota","A quantum hardware engineer at IBM and graduate researcher at the Nano Magnetism & Quantum Spintronics Lab, University of Minnesota-Twin Cities. Onri shares runnable notebooks that make hardware-level quantum concepts visible.",[978,981],{"label":979,"href":980},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":982,"href":478},"Original notebook ↗",{"username":467,"name":974,"role":984},"Quantum Hardware Engineer, IBM","One entangled state, eight ways to see it. A runnable notebook that turns the same 2-qubit circuit into Bloch spheres, steering ellipsoids, Wigner functions, and more.","One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, Wigner functions, and more.","Deep dive · Visualization","article",{},"\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.png","\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization","2026-07-15","6 min read",[],[996,1003,1010],{"username":997,"project":998,"title":999,"category":1000,"thumb":1001,"to":1002},"q-inho","qave","QAVE","Visualization","\u002F_content\u002Fimages\u002Fqave\u002Fthumbnail.webp","\u002Fexplore\u002Fqave",{"username":1004,"project":1005,"title":1006,"category":1007,"thumb":1008,"to":1009},"AmberPincar","quantum-garden","Quantum Garden","Art","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-garden",{"username":1011,"project":1012,"title":1013,"category":1007,"thumb":1014},"xinyi","quantum-butterfly-field","Quantum Butterfly Field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fthumbnail.webp",{"description":1016,"title":465},"One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, and more.","blog\u002Fexpert-notes\u002F2-qubit-state-visualization",[1019,143,502],"visualization","hbpIuxS-SqH0GmOc1n1ds2B-L8OsTlp8DNHQ4vd3Ylw",{"id":1022,"title":1023,"authors":1024,"body":1026,"breadcrumb":41633,"builders":41635,"byline":41643,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":116,"description":41645,"draft":125,"extension":126,"eyebrow":41646,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":988,"lessonCount":116,"meta":41647,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":41648,"publishDate":992,"readingTime":41649,"related":41650,"relatedProjects":116,"seo":41651,"stem":41652,"tags":41653,"track":116,"trackName":116,"__hash__":41656},"blog\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization.md","Krylov Quantum Diagonalization of Lattice Hamiltonians",[1025],"mirko-amico",{"type":10,"value":1027,"toc":41604},[1028,1039,1044,1059,1063,1066,1094,1098,1101,1117,1121,2508,2512,2517,2781,3092,3759,4079,4082,4086,4390,4651,4922,5126,5905,6142,6145,6383,6705,6712,7659,7664,8150,8528,8535,8539,8637,8973,8979,8982,9030,9034,9116,9182,9186,9190,9298,9422,9428,9432,9438,10709,11538,12088,12091,13079,13471,13718,14128,14134,14137,14140,14207,14213,14217,14221,14224,14230,14874,17641,17849,19068,20302,20347,20351,20899,21331,21334,21338,21428,21431,21556,21559,21617,21842,21845,21900,21906,21909,21913,21916,22023,22027,22030,22419,22422,22637,22643,22647,23115,23119,23174,23504,23510,23602,23975,23979,24004,24008,24048,24138,24169,24495,24508,24571,24910,24922,24986,25158,25242,25335,25341,25344,25361,25367,25616,25620,25623,25997,26123,26719,26751,27216,27298,27304,27369,28086,28189,28452,28706,29037,29220,29562,30302,30616,32231,32263,33759,34236,34930,35713,36047,36374,36436,36445,36449,36648,37006,37009,37537,37540,37840,37896,38160,38653,38912,39854,40858,41298,41553,41557,41566,41571,41576,41581,41585,41588,41594,41596,41601],[18,1029,1030],{},[1031,1032,1033,1034,1038],"em",{},"Republished with the author's permission from ",[49,1035,1037],{"href":1036},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fen\u002Ftutorials\u002Fkrylov-quantum-diagonalization","the original tutorial",". The text, code, and figures below are Mirko's, as published, lightly reformatted for this page.",[18,1040,1041],{},[1031,1042,1043],{},"Usage estimate: 20 minutes on a Heron r2 (NOTE: This is an estimate only. Your runtime might vary.)",[498,1045,1047],{"className":500,"code":1046,"language":502,"meta":104,"style":104},"# This cell is hidden from users – it disables some lint rules\n# ruff: noqa: E402 E722 F601\n",[504,1048,1049,1054],{"__ignoreMap":104},[507,1050,1051],{"class":509,"line":510},[507,1052,1053],{"class":562},"# This cell is hidden from users – it disables some lint rules\n",[507,1055,1056],{"class":509,"line":105},[507,1057,1058],{"class":562},"# ruff: noqa: E402 E722 F601\n",[13,1060,1062],{"id":1061},"background","Background",[18,1064,1065],{},"This tutorial demonstrates how to implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns. You will first learn about the theory behind the algorithm and then see a demonstration of its execution on a QPU.",[18,1067,1068,1069,1073,1074,1078,1079,1083,1084,1088,1089,1093],{},"Across disciplines, we're interested in learning ground state properties of quantum systems. Examples include understanding the fundamental nature of particles and forces, predicting and understanding the behavior of complex materials and understanding bio-chemical interactions and reactions. Because of the exponential growth of the Hilbert space and the correlation that arise in entangled systems, classical algorithm struggle to solve this problem for quantum systems of increasing size. At one end of the spectrum is the existing approach that takes advantage of the quantum hardware focus on variational quantum methods (for example, ",[49,1070,1072],{"href":1071},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Ftutorials\u002Fspin-chain-vqe","variational quantum eigensolver","). These techniques face challenges with current devices because of the high number of function calls required in the optimization process, which adds a large overhead in resources once advanced error mitigation techniques are introduced, thus limiting their efficacy to small systems. At the other end of the spectrum, there are fault-tolerant quantum methods with performance guarantees (for example, ",[49,1075,1077],{"href":1076},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fquant-ph\u002F0604193","quantum phase estimation","), which require deep circuits that can be executed only on a fault-tolerant device. For these reasons, we introduce here a quantum algorithm based on subspace methods (as described in this ",[49,1080,1082],{"href":1081},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00178","review paper","), the Krylov quantum diagonalization (KQD) algorithm. This algorithm performs well at large scale ",[49,1085,1087],{"href":1086},"#references","[1]"," on existing quantum hardware, shares similar ",[49,1090,1092],{"href":1091},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.07492","performance guarantees"," as phase estimation, is compatible with advanced error mitigation techniques, and could provide results that are classically inaccessible.",[13,1095,1097],{"id":1096},"requirements","Requirements",[18,1099,1100],{},"Before starting this tutorial, be sure you have the following installed:",[42,1102,1103,1110],{},[45,1104,1105,1106,1109],{},"Qiskit SDK v2.0 or later, with ",[49,1107,1019],{"href":1108},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fvisualization"," support",[45,1111,1112,1113,1116],{},"Qiskit Runtime v0.22 or later ( ",[504,1114,1115],{},"pip install qiskit-ibm-runtime"," )",[13,1118,1120],{"id":1119},"setup","Setup",[498,1122,1124],{"className":500,"code":1123,"language":502,"meta":104,"style":104},"import numpy as np\nimport scipy as sp\nimport matplotlib.pylab as plt\nfrom typing import Union, List\nimport itertools as it\nimport copy\nfrom sympy import Matrix\nimport warnings\n\nwarnings.filterwarnings(\"ignore\")\n\nfrom qiskit.quantum_info import SparsePauliOp, Pauli, StabilizerState\nfrom qiskit.circuit import Parameter, IfElseOp\nfrom qiskit import QuantumCircuit, QuantumRegister\nfrom qiskit.circuit.library import PauliEvolutionGate\nfrom qiskit.synthesis import LieTrotter\nfrom qiskit.transpiler import Target, CouplingMap\nfrom qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n\n\nfrom qiskit_ibm_runtime import (\n    QiskitRuntimeService,\n    EstimatorV2 as Estimator,\n)\n\n\ndef solve_regularized_gen_eig(\n    h: np.ndarray,\n    s: np.ndarray,\n    threshold: float,\n    k: int = 1,\n    return_dimn: bool = False,\n) -> Union[float, List[float]]:\n    \"\"\"\n    Method for solving the generalized eigenvalue problem with regularization\n\n    Args:\n        h (numpy.ndarray):\n            The effective representation of the matrix in the Krylov subspace\n        s (numpy.ndarray):\n            The matrix of overlaps between vectors of the Krylov subspace\n        threshold (float):\n            Cut-off value for the eigenvalue of s\n        k (int):\n            Number of eigenvalues to return\n        return_dimn (bool):\n            Whether to return the size of the regularized subspace\n\n    Returns:\n        lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n\n\n    \"\"\"\n    s_vals, s_vecs = sp.linalg.eigh(s)\n    s_vecs = s_vecs.T\n    good_vecs = np.array(\n        [vec for val, vec in zip(s_vals, s_vecs) if val > threshold]\n    )\n    h_reg = good_vecs.conj() @ h @ good_vecs.T\n    s_reg = good_vecs.conj() @ s @ good_vecs.T\n    if k == 1:\n        if return_dimn:\n            return sp.linalg.eigh(h_reg, s_reg)[0][0], len(good_vecs)\n        else:\n            return sp.linalg.eigh(h_reg, s_reg)[0][0]\n    else:\n        if return_dimn:\n            return sp.linalg.eigh(h_reg, s_reg)[0][:k], len(good_vecs)\n        else:\n            return sp.linalg.eigh(h_reg, s_reg)[0][:k]\n\n\ndef single_particle_gs(H_op, n_qubits):\n    \"\"\"\n    Find the ground state of the single particle(excitation) sector\n    \"\"\"\n    H_x = []\n    for p, coeff in H_op.to_list():\n        H_x.append(set([i for i, v in enumerate(Pauli(p).x) if v]))\n\n    H_z = []\n    for p, coeff in H_op.to_list():\n        H_z.append(set([i for i, v in enumerate(Pauli(p).z) if v]))\n\n    H_c = H_op.coeffs\n\n    print(\"n_sys_qubits\", n_qubits)\n\n    n_exc = 1\n    sub_dimn = int(sp.special.comb(n_qubits + 1, n_exc))\n    print(\"n_exc\", n_exc, \", subspace dimension\", sub_dimn)\n\n    few_particle_H = np.zeros((sub_dimn, sub_dimn), dtype=complex)\n\n    sparse_vecs = [\n        set(vec) for vec in it.combinations(range(n_qubits + 1), r=n_exc)\n    ]  # list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n\n    m = 0\n    for i, i_set in enumerate(sparse_vecs):\n        for j, j_set in enumerate(sparse_vecs):\n            m += 1\n\n            if len(i_set.symmetric_difference(j_set)) \u003C= 2:\n                for p_x, p_z, coeff in zip(H_x, H_z, H_c):\n                    if i_set.symmetric_difference(j_set) == p_x:\n                        sgn = ((-1j) ** len(p_x.intersection(p_z))) * (\n                            (-1) ** len(i_set.intersection(p_z))\n                        )\n                    else:\n                        sgn = 0\n\n                    few_particle_H[i, j] += sgn * coeff\n\n    gs_en = min(np.linalg.eigvalsh(few_particle_H))\n    print(\"single particle ground state energy: \", gs_en)\n    return gs_en\n",[504,1125,1126,1136,1148,1160,1172,1184,1191,1203,1210,1214,1229,1233,1244,1256,1267,1279,1291,1303,1315,1319,1323,1335,1340,1350,1355,1360,1365,1377,1387,1395,1410,1429,1447,1463,1469,1475,1480,1486,1492,1498,1504,1510,1516,1522,1528,1534,1540,1546,1551,1557,1563,1568,1573,1578,1595,1606,1622,1655,1661,1689,1712,1729,1738,1767,1775,1795,1803,1810,1830,1837,1853,1858,1863,1884,1889,1895,1900,1911,1931,1971,1976,1986,2001,2034,2039,2050,2055,2069,2074,2085,2113,2132,2137,2162,2167,2178,2222,2231,2236,2247,2262,2277,2288,2293,2319,2335,2354,2394,2417,2423,2431,2440,2445,2461,2466,2486,2499],{"__ignoreMap":104},[507,1127,1128,1130,1132,1134],{"class":509,"line":510},[507,1129,514],{"class":513},[507,1131,518],{"class":517},[507,1133,521],{"class":513},[507,1135,524],{"class":517},[507,1137,1138,1140,1143,1145],{"class":509,"line":105},[507,1139,514],{"class":513},[507,1141,1142],{"class":517}," scipy ",[507,1144,521],{"class":513},[507,1146,1147],{"class":517}," sp\n",[507,1149,1150,1152,1155,1157],{"class":509,"line":540},[507,1151,514],{"class":513},[507,1153,1154],{"class":517}," matplotlib.pylab ",[507,1156,521],{"class":513},[507,1158,1159],{"class":517}," plt\n",[507,1161,1162,1164,1167,1169],{"class":509,"line":553},[507,1163,529],{"class":513},[507,1165,1166],{"class":517}," typing ",[507,1168,514],{"class":513},[507,1170,1171],{"class":517}," Union, List\n",[507,1173,1174,1176,1179,1181],{"class":509,"line":559},[507,1175,514],{"class":513},[507,1177,1178],{"class":517}," itertools ",[507,1180,521],{"class":513},[507,1182,1183],{"class":517}," it\n",[507,1185,1186,1188],{"class":509,"line":566},[507,1187,514],{"class":513},[507,1189,1190],{"class":517}," copy\n",[507,1192,1193,1195,1198,1200],{"class":509,"line":590},[507,1194,529],{"class":513},[507,1196,1197],{"class":517}," sympy ",[507,1199,514],{"class":513},[507,1201,1202],{"class":517}," Matrix\n",[507,1204,1205,1207],{"class":509,"line":610},[507,1206,514],{"class":513},[507,1208,1209],{"class":517}," warnings\n",[507,1211,1212],{"class":509,"line":634},[507,1213,556],{"emptyLinePlaceholder":133},[507,1215,1216,1219,1222,1224,1227],{"class":509,"line":661},[507,1217,1218],{"class":517},"warnings.",[507,1220,1221],{"class":576},"filterwarnings",[507,1223,580],{"class":517},[507,1225,1226],{"class":730},"\"ignore\"",[507,1228,587],{"class":517},[507,1230,1231],{"class":509,"line":678},[507,1232,556],{"emptyLinePlaceholder":133},[507,1234,1235,1237,1239,1241],{"class":509,"line":683},[507,1236,529],{"class":513},[507,1238,545],{"class":517},[507,1240,514],{"class":513},[507,1242,1243],{"class":517}," SparsePauliOp, Pauli, StabilizerState\n",[507,1245,1246,1248,1251,1253],{"class":509,"line":697},[507,1247,529],{"class":513},[507,1249,1250],{"class":517}," qiskit.circuit ",[507,1252,514],{"class":513},[507,1254,1255],{"class":517}," Parameter, IfElseOp\n",[507,1257,1258,1260,1262,1264],{"class":509,"line":710},[507,1259,529],{"class":513},[507,1261,532],{"class":517},[507,1263,514],{"class":513},[507,1265,1266],{"class":517}," QuantumCircuit, QuantumRegister\n",[507,1268,1269,1271,1274,1276],{"class":509,"line":715},[507,1270,529],{"class":513},[507,1272,1273],{"class":517}," qiskit.circuit.library ",[507,1275,514],{"class":513},[507,1277,1278],{"class":517}," PauliEvolutionGate\n",[507,1280,1281,1283,1286,1288],{"class":509,"line":721},[507,1282,529],{"class":513},[507,1284,1285],{"class":517}," qiskit.synthesis ",[507,1287,514],{"class":513},[507,1289,1290],{"class":517}," LieTrotter\n",[507,1292,1293,1295,1298,1300],{"class":509,"line":736},[507,1294,529],{"class":513},[507,1296,1297],{"class":517}," qiskit.transpiler ",[507,1299,514],{"class":513},[507,1301,1302],{"class":517}," Target, CouplingMap\n",[507,1304,1305,1307,1310,1312],{"class":509,"line":748},[507,1306,529],{"class":513},[507,1308,1309],{"class":517}," qiskit.transpiler.preset_passmanagers ",[507,1311,514],{"class":513},[507,1313,1314],{"class":517}," generate_preset_pass_manager\n",[507,1316,1317],{"class":509,"line":761},[507,1318,556],{"emptyLinePlaceholder":133},[507,1320,1321],{"class":509,"line":775},[507,1322,556],{"emptyLinePlaceholder":133},[507,1324,1325,1327,1330,1332],{"class":509,"line":784},[507,1326,529],{"class":513},[507,1328,1329],{"class":517}," qiskit_ibm_runtime ",[507,1331,514],{"class":513},[507,1333,1334],{"class":517}," (\n",[507,1336,1337],{"class":509,"line":796},[507,1338,1339],{"class":517},"    QiskitRuntimeService,\n",[507,1341,1342,1345,1347],{"class":509,"line":809},[507,1343,1344],{"class":517},"    EstimatorV2 ",[507,1346,521],{"class":513},[507,1348,1349],{"class":517}," Estimator,\n",[507,1351,1353],{"class":509,"line":1352},24,[507,1354,587],{"class":517},[507,1356,1358],{"class":509,"line":1357},25,[507,1359,556],{"emptyLinePlaceholder":133},[507,1361,1363],{"class":509,"line":1362},26,[507,1364,556],{"emptyLinePlaceholder":133},[507,1366,1368,1371,1374],{"class":509,"line":1367},27,[507,1369,1370],{"class":513},"def",[507,1372,1373],{"class":576}," solve_regularized_gen_eig",[507,1375,1376],{"class":517},"(\n",[507,1378,1380,1384],{"class":509,"line":1379},28,[507,1381,1383],{"class":1382},"sb9H8","    h",[507,1385,1386],{"class":517},": np.ndarray,\n",[507,1388,1390,1393],{"class":509,"line":1389},29,[507,1391,1392],{"class":1382},"    s",[507,1394,1386],{"class":517},[507,1396,1398,1401,1404,1407],{"class":509,"line":1397},30,[507,1399,1400],{"class":1382},"    threshold",[507,1402,1403],{"class":517},": ",[507,1405,1406],{"class":572},"float",[507,1408,1409],{"class":517},",\n",[507,1411,1413,1416,1418,1421,1424,1427],{"class":509,"line":1412},31,[507,1414,1415],{"class":1382},"    k",[507,1417,1403],{"class":517},[507,1419,1420],{"class":572},"int",[507,1422,1423],{"class":572}," =",[507,1425,1426],{"class":583}," 1",[507,1428,1409],{"class":517},[507,1430,1432,1435,1437,1440,1442,1445],{"class":509,"line":1431},32,[507,1433,1434],{"class":1382},"    return_dimn",[507,1436,1403],{"class":517},[507,1438,1439],{"class":572},"bool",[507,1441,1423],{"class":572},[507,1443,1444],{"class":583}," False",[507,1446,1409],{"class":517},[507,1448,1450,1453,1455,1458,1460],{"class":509,"line":1449},33,[507,1451,1452],{"class":517},") -> Union[",[507,1454,1406],{"class":572},[507,1456,1457],{"class":517},", List[",[507,1459,1406],{"class":572},[507,1461,1462],{"class":517},"]]:\n",[507,1464,1466],{"class":509,"line":1465},34,[507,1467,1468],{"class":730},"    \"\"\"\n",[507,1470,1472],{"class":509,"line":1471},35,[507,1473,1474],{"class":730},"    Method for solving the generalized eigenvalue problem with regularization\n",[507,1476,1478],{"class":509,"line":1477},36,[507,1479,556],{"emptyLinePlaceholder":133},[507,1481,1483],{"class":509,"line":1482},37,[507,1484,1485],{"class":730},"    Args:\n",[507,1487,1489],{"class":509,"line":1488},38,[507,1490,1491],{"class":730},"        h (numpy.ndarray):\n",[507,1493,1495],{"class":509,"line":1494},39,[507,1496,1497],{"class":730},"            The effective representation of the matrix in the Krylov subspace\n",[507,1499,1501],{"class":509,"line":1500},40,[507,1502,1503],{"class":730},"        s (numpy.ndarray):\n",[507,1505,1507],{"class":509,"line":1506},41,[507,1508,1509],{"class":730},"            The matrix of overlaps between vectors of the Krylov subspace\n",[507,1511,1513],{"class":509,"line":1512},42,[507,1514,1515],{"class":730},"        threshold (float):\n",[507,1517,1519],{"class":509,"line":1518},43,[507,1520,1521],{"class":730},"            Cut-off value for the eigenvalue of s\n",[507,1523,1525],{"class":509,"line":1524},44,[507,1526,1527],{"class":730},"        k (int):\n",[507,1529,1531],{"class":509,"line":1530},45,[507,1532,1533],{"class":730},"            Number of eigenvalues to return\n",[507,1535,1537],{"class":509,"line":1536},46,[507,1538,1539],{"class":730},"        return_dimn (bool):\n",[507,1541,1543],{"class":509,"line":1542},47,[507,1544,1545],{"class":730},"            Whether to return the size of the regularized subspace\n",[507,1547,1549],{"class":509,"line":1548},48,[507,1550,556],{"emptyLinePlaceholder":133},[507,1552,1554],{"class":509,"line":1553},49,[507,1555,1556],{"class":730},"    Returns:\n",[507,1558,1560],{"class":509,"line":1559},50,[507,1561,1562],{"class":730},"        lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n",[507,1564,1566],{"class":509,"line":1565},51,[507,1567,556],{"emptyLinePlaceholder":133},[507,1569,1571],{"class":509,"line":1570},52,[507,1572,556],{"emptyLinePlaceholder":133},[507,1574,1576],{"class":509,"line":1575},53,[507,1577,1468],{"class":730},[507,1579,1581,1584,1586,1589,1592],{"class":509,"line":1580},54,[507,1582,1583],{"class":517},"    s_vals, s_vecs ",[507,1585,573],{"class":572},[507,1587,1588],{"class":517}," sp.linalg.",[507,1590,1591],{"class":576},"eigh",[507,1593,1594],{"class":517},"(s)\n",[507,1596,1598,1601,1603],{"class":509,"line":1597},55,[507,1599,1600],{"class":517},"    s_vecs ",[507,1602,573],{"class":572},[507,1604,1605],{"class":517}," s_vecs.T\n",[507,1607,1609,1612,1614,1617,1620],{"class":509,"line":1608},56,[507,1610,1611],{"class":517},"    good_vecs ",[507,1613,573],{"class":572},[507,1615,1616],{"class":517}," np.",[507,1618,1619],{"class":576},"array",[507,1621,1376],{"class":517},[507,1623,1625,1628,1631,1634,1637,1640,1643,1646,1649,1652],{"class":509,"line":1624},57,[507,1626,1627],{"class":517},"        [vec ",[507,1629,1630],{"class":513},"for",[507,1632,1633],{"class":517}," val, vec ",[507,1635,1636],{"class":513},"in",[507,1638,1639],{"class":572}," zip",[507,1641,1642],{"class":517},"(s_vals, s_vecs) ",[507,1644,1645],{"class":513},"if",[507,1647,1648],{"class":517}," val ",[507,1650,1651],{"class":572},">",[507,1653,1654],{"class":517}," threshold]\n",[507,1656,1658],{"class":509,"line":1657},58,[507,1659,1660],{"class":517},"    )\n",[507,1662,1664,1667,1669,1672,1675,1678,1681,1684,1686],{"class":509,"line":1663},59,[507,1665,1666],{"class":517},"    h_reg ",[507,1668,573],{"class":572},[507,1670,1671],{"class":517}," good_vecs.",[507,1673,1674],{"class":576},"conj",[507,1676,1677],{"class":517},"() ",[507,1679,1680],{"class":572},"@",[507,1682,1683],{"class":517}," h ",[507,1685,1680],{"class":572},[507,1687,1688],{"class":517}," good_vecs.T\n",[507,1690,1692,1695,1697,1699,1701,1703,1705,1708,1710],{"class":509,"line":1691},60,[507,1693,1694],{"class":517},"    s_reg ",[507,1696,573],{"class":572},[507,1698,1671],{"class":517},[507,1700,1674],{"class":576},[507,1702,1677],{"class":517},[507,1704,1680],{"class":572},[507,1706,1707],{"class":517}," s ",[507,1709,1680],{"class":572},[507,1711,1688],{"class":517},[507,1713,1715,1718,1721,1724,1726],{"class":509,"line":1714},61,[507,1716,1717],{"class":513},"    if",[507,1719,1720],{"class":517}," k ",[507,1722,1723],{"class":572},"==",[507,1725,1426],{"class":583},[507,1727,1728],{"class":517},":\n",[507,1730,1732,1735],{"class":509,"line":1731},62,[507,1733,1734],{"class":513},"        if",[507,1736,1737],{"class":517}," return_dimn:\n",[507,1739,1741,1744,1746,1748,1751,1753,1756,1758,1761,1764],{"class":509,"line":1740},63,[507,1742,1743],{"class":513},"            return",[507,1745,1588],{"class":517},[507,1747,1591],{"class":576},[507,1749,1750],{"class":517},"(h_reg, s_reg)[",[507,1752,601],{"class":583},[507,1754,1755],{"class":517},"][",[507,1757,601],{"class":583},[507,1759,1760],{"class":517},"], ",[507,1762,1763],{"class":572},"len",[507,1765,1766],{"class":517},"(good_vecs)\n",[507,1768,1770,1773],{"class":509,"line":1769},64,[507,1771,1772],{"class":513},"        else",[507,1774,1728],{"class":517},[507,1776,1778,1780,1782,1784,1786,1788,1790,1792],{"class":509,"line":1777},65,[507,1779,1743],{"class":513},[507,1781,1588],{"class":517},[507,1783,1591],{"class":576},[507,1785,1750],{"class":517},[507,1787,601],{"class":583},[507,1789,1755],{"class":517},[507,1791,601],{"class":583},[507,1793,1794],{"class":517},"]\n",[507,1796,1798,1801],{"class":509,"line":1797},66,[507,1799,1800],{"class":513},"    else",[507,1802,1728],{"class":517},[507,1804,1806,1808],{"class":509,"line":1805},67,[507,1807,1734],{"class":513},[507,1809,1737],{"class":517},[507,1811,1813,1815,1817,1819,1821,1823,1826,1828],{"class":509,"line":1812},68,[507,1814,1743],{"class":513},[507,1816,1588],{"class":517},[507,1818,1591],{"class":576},[507,1820,1750],{"class":517},[507,1822,601],{"class":583},[507,1824,1825],{"class":517},"][:k], ",[507,1827,1763],{"class":572},[507,1829,1766],{"class":517},[507,1831,1833,1835],{"class":509,"line":1832},69,[507,1834,1772],{"class":513},[507,1836,1728],{"class":517},[507,1838,1840,1842,1844,1846,1848,1850],{"class":509,"line":1839},70,[507,1841,1743],{"class":513},[507,1843,1588],{"class":517},[507,1845,1591],{"class":576},[507,1847,1750],{"class":517},[507,1849,601],{"class":583},[507,1851,1852],{"class":517},"][:k]\n",[507,1854,1856],{"class":509,"line":1855},71,[507,1857,556],{"emptyLinePlaceholder":133},[507,1859,1861],{"class":509,"line":1860},72,[507,1862,556],{"emptyLinePlaceholder":133},[507,1864,1866,1868,1871,1873,1876,1878,1881],{"class":509,"line":1865},73,[507,1867,1370],{"class":513},[507,1869,1870],{"class":576}," single_particle_gs",[507,1872,580],{"class":517},[507,1874,1875],{"class":1382},"H_op",[507,1877,622],{"class":517},[507,1879,1880],{"class":1382},"n_qubits",[507,1882,1883],{"class":517},"):\n",[507,1885,1887],{"class":509,"line":1886},74,[507,1888,1468],{"class":730},[507,1890,1892],{"class":509,"line":1891},75,[507,1893,1894],{"class":730},"    Find the ground state of the single particle(excitation) sector\n",[507,1896,1898],{"class":509,"line":1897},76,[507,1899,1468],{"class":730},[507,1901,1903,1906,1908],{"class":509,"line":1902},77,[507,1904,1905],{"class":517},"    H_x ",[507,1907,573],{"class":572},[507,1909,1910],{"class":517}," []\n",[507,1912,1914,1917,1920,1922,1925,1928],{"class":509,"line":1913},78,[507,1915,1916],{"class":513},"    for",[507,1918,1919],{"class":517}," p, coeff ",[507,1921,1636],{"class":513},[507,1923,1924],{"class":517}," H_op.",[507,1926,1927],{"class":576},"to_list",[507,1929,1930],{"class":517},"():\n",[507,1932,1934,1937,1940,1942,1945,1948,1950,1953,1955,1958,1960,1963,1966,1968],{"class":509,"line":1933},79,[507,1935,1936],{"class":517},"        H_x.",[507,1938,1939],{"class":576},"append",[507,1941,580],{"class":517},[507,1943,1944],{"class":572},"set",[507,1946,1947],{"class":517},"([i ",[507,1949,1630],{"class":513},[507,1951,1952],{"class":517}," i, v ",[507,1954,1636],{"class":513},[507,1956,1957],{"class":572}," enumerate",[507,1959,580],{"class":517},[507,1961,1962],{"class":576},"Pauli",[507,1964,1965],{"class":517},"(p).x) ",[507,1967,1645],{"class":513},[507,1969,1970],{"class":517}," v]))\n",[507,1972,1974],{"class":509,"line":1973},80,[507,1975,556],{"emptyLinePlaceholder":133},[507,1977,1979,1982,1984],{"class":509,"line":1978},81,[507,1980,1981],{"class":517},"    H_z ",[507,1983,573],{"class":572},[507,1985,1910],{"class":517},[507,1987,1989,1991,1993,1995,1997,1999],{"class":509,"line":1988},82,[507,1990,1916],{"class":513},[507,1992,1919],{"class":517},[507,1994,1636],{"class":513},[507,1996,1924],{"class":517},[507,1998,1927],{"class":576},[507,2000,1930],{"class":517},[507,2002,2004,2007,2009,2011,2013,2015,2017,2019,2021,2023,2025,2027,2030,2032],{"class":509,"line":2003},83,[507,2005,2006],{"class":517},"        H_z.",[507,2008,1939],{"class":576},[507,2010,580],{"class":517},[507,2012,1944],{"class":572},[507,2014,1947],{"class":517},[507,2016,1630],{"class":513},[507,2018,1952],{"class":517},[507,2020,1636],{"class":513},[507,2022,1957],{"class":572},[507,2024,580],{"class":517},[507,2026,1962],{"class":576},[507,2028,2029],{"class":517},"(p).z) ",[507,2031,1645],{"class":513},[507,2033,1970],{"class":517},[507,2035,2037],{"class":509,"line":2036},84,[507,2038,556],{"emptyLinePlaceholder":133},[507,2040,2042,2045,2047],{"class":509,"line":2041},85,[507,2043,2044],{"class":517},"    H_c ",[507,2046,573],{"class":572},[507,2048,2049],{"class":517}," H_op.coeffs\n",[507,2051,2053],{"class":509,"line":2052},86,[507,2054,556],{"emptyLinePlaceholder":133},[507,2056,2058,2061,2063,2066],{"class":509,"line":2057},87,[507,2059,2060],{"class":572},"    print",[507,2062,580],{"class":517},[507,2064,2065],{"class":730},"\"n_sys_qubits\"",[507,2067,2068],{"class":517},", n_qubits)\n",[507,2070,2072],{"class":509,"line":2071},88,[507,2073,556],{"emptyLinePlaceholder":133},[507,2075,2077,2080,2082],{"class":509,"line":2076},89,[507,2078,2079],{"class":517},"    n_exc ",[507,2081,573],{"class":572},[507,2083,2084],{"class":583}," 1\n",[507,2086,2088,2091,2093,2096,2099,2102,2105,2108,2110],{"class":509,"line":2087},90,[507,2089,2090],{"class":517},"    sub_dimn ",[507,2092,573],{"class":572},[507,2094,2095],{"class":572}," int",[507,2097,2098],{"class":517},"(sp.special.",[507,2100,2101],{"class":576},"comb",[507,2103,2104],{"class":517},"(n_qubits ",[507,2106,2107],{"class":572},"+",[507,2109,1426],{"class":583},[507,2111,2112],{"class":517},", n_exc))\n",[507,2114,2116,2118,2120,2123,2126,2129],{"class":509,"line":2115},91,[507,2117,2060],{"class":572},[507,2119,580],{"class":517},[507,2121,2122],{"class":730},"\"n_exc\"",[507,2124,2125],{"class":517},", n_exc, ",[507,2127,2128],{"class":730},"\", subspace dimension\"",[507,2130,2131],{"class":517},", sub_dimn)\n",[507,2133,2135],{"class":509,"line":2134},92,[507,2136,556],{"emptyLinePlaceholder":133},[507,2138,2140,2143,2145,2147,2150,2153,2157,2160],{"class":509,"line":2139},93,[507,2141,2142],{"class":517},"    few_particle_H ",[507,2144,573],{"class":572},[507,2146,1616],{"class":517},[507,2148,2149],{"class":576},"zeros",[507,2151,2152],{"class":517},"((sub_dimn, sub_dimn), ",[507,2154,2156],{"class":2155},"s_ZVi","dtype",[507,2158,2159],{"class":572},"=complex",[507,2161,587],{"class":517},[507,2163,2165],{"class":509,"line":2164},94,[507,2166,556],{"emptyLinePlaceholder":133},[507,2168,2170,2173,2175],{"class":509,"line":2169},95,[507,2171,2172],{"class":517},"    sparse_vecs ",[507,2174,573],{"class":572},[507,2176,2177],{"class":517}," [\n",[507,2179,2181,2184,2187,2189,2192,2194,2197,2200,2202,2205,2207,2209,2211,2214,2217,2219],{"class":509,"line":2180},96,[507,2182,2183],{"class":572},"        set",[507,2185,2186],{"class":517},"(vec) ",[507,2188,1630],{"class":513},[507,2190,2191],{"class":517}," vec ",[507,2193,1636],{"class":513},[507,2195,2196],{"class":517}," it.",[507,2198,2199],{"class":576},"combinations",[507,2201,580],{"class":517},[507,2203,2204],{"class":572},"range",[507,2206,2104],{"class":517},[507,2208,2107],{"class":572},[507,2210,1426],{"class":583},[507,2212,2213],{"class":517},"), ",[507,2215,2216],{"class":2155},"r",[507,2218,573],{"class":572},[507,2220,2221],{"class":517},"n_exc)\n",[507,2223,2225,2228],{"class":509,"line":2224},97,[507,2226,2227],{"class":517},"    ]  ",[507,2229,2230],{"class":562},"# list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n",[507,2232,2234],{"class":509,"line":2233},98,[507,2235,556],{"emptyLinePlaceholder":133},[507,2237,2239,2242,2244],{"class":509,"line":2238},99,[507,2240,2241],{"class":517},"    m ",[507,2243,573],{"class":572},[507,2245,2246],{"class":583}," 0\n",[507,2248,2250,2252,2255,2257,2259],{"class":509,"line":2249},100,[507,2251,1916],{"class":513},[507,2253,2254],{"class":517}," i, i_set ",[507,2256,1636],{"class":513},[507,2258,1957],{"class":572},[507,2260,2261],{"class":517},"(sparse_vecs):\n",[507,2263,2265,2268,2271,2273,2275],{"class":509,"line":2264},101,[507,2266,2267],{"class":513},"        for",[507,2269,2270],{"class":517}," j, j_set ",[507,2272,1636],{"class":513},[507,2274,1957],{"class":572},[507,2276,2261],{"class":517},[507,2278,2280,2283,2286],{"class":509,"line":2279},102,[507,2281,2282],{"class":517},"            m ",[507,2284,2285],{"class":572},"+=",[507,2287,2084],{"class":583},[507,2289,2291],{"class":509,"line":2290},103,[507,2292,556],{"emptyLinePlaceholder":133},[507,2294,2296,2299,2302,2305,2308,2311,2314,2317],{"class":509,"line":2295},104,[507,2297,2298],{"class":513},"            if",[507,2300,2301],{"class":572}," len",[507,2303,2304],{"class":517},"(i_set.",[507,2306,2307],{"class":576},"symmetric_difference",[507,2309,2310],{"class":517},"(j_set)) ",[507,2312,2313],{"class":572},"\u003C=",[507,2315,2316],{"class":583}," 2",[507,2318,1728],{"class":517},[507,2320,2322,2325,2328,2330,2332],{"class":509,"line":2321},105,[507,2323,2324],{"class":513},"                for",[507,2326,2327],{"class":517}," p_x, p_z, coeff ",[507,2329,1636],{"class":513},[507,2331,1639],{"class":572},[507,2333,2334],{"class":517},"(H_x, H_z, H_c):\n",[507,2336,2338,2341,2344,2346,2349,2351],{"class":509,"line":2337},106,[507,2339,2340],{"class":513},"                    if",[507,2342,2343],{"class":517}," i_set.",[507,2345,2307],{"class":576},[507,2347,2348],{"class":517},"(j_set) ",[507,2350,1723],{"class":572},[507,2352,2353],{"class":517}," p_x:\n",[507,2355,2357,2360,2362,2365,2368,2370,2373,2375,2378,2380,2383,2386,2389,2392],{"class":509,"line":2356},107,[507,2358,2359],{"class":517},"                        sgn ",[507,2361,573],{"class":572},[507,2363,2364],{"class":517}," ((",[507,2366,2367],{"class":572},"-",[507,2369,625],{"class":583},[507,2371,2372],{"class":513},"j",[507,2374,655],{"class":517},[507,2376,2377],{"class":572},"**",[507,2379,2301],{"class":572},[507,2381,2382],{"class":517},"(p_x.",[507,2384,2385],{"class":576},"intersection",[507,2387,2388],{"class":517},"(p_z))) ",[507,2390,2391],{"class":572},"*",[507,2393,1334],{"class":517},[507,2395,2397,2400,2402,2404,2406,2408,2410,2412,2414],{"class":509,"line":2396},108,[507,2398,2399],{"class":517},"                            (",[507,2401,2367],{"class":572},[507,2403,625],{"class":583},[507,2405,655],{"class":517},[507,2407,2377],{"class":572},[507,2409,2301],{"class":572},[507,2411,2304],{"class":517},[507,2413,2385],{"class":576},[507,2415,2416],{"class":517},"(p_z))\n",[507,2418,2420],{"class":509,"line":2419},109,[507,2421,2422],{"class":517},"                        )\n",[507,2424,2426,2429],{"class":509,"line":2425},110,[507,2427,2428],{"class":513},"                    else",[507,2430,1728],{"class":517},[507,2432,2434,2436,2438],{"class":509,"line":2433},111,[507,2435,2359],{"class":517},[507,2437,573],{"class":572},[507,2439,2246],{"class":583},[507,2441,2443],{"class":509,"line":2442},112,[507,2444,556],{"emptyLinePlaceholder":133},[507,2446,2448,2451,2453,2456,2458],{"class":509,"line":2447},113,[507,2449,2450],{"class":517},"                    few_particle_H[i, j] ",[507,2452,2285],{"class":572},[507,2454,2455],{"class":517}," sgn ",[507,2457,2391],{"class":572},[507,2459,2460],{"class":517}," coeff\n",[507,2462,2464],{"class":509,"line":2463},114,[507,2465,556],{"emptyLinePlaceholder":133},[507,2467,2469,2472,2474,2477,2480,2483],{"class":509,"line":2468},115,[507,2470,2471],{"class":517},"    gs_en ",[507,2473,573],{"class":572},[507,2475,2476],{"class":572}," min",[507,2478,2479],{"class":517},"(np.linalg.",[507,2481,2482],{"class":576},"eigvalsh",[507,2484,2485],{"class":517},"(few_particle_H))\n",[507,2487,2489,2491,2493,2496],{"class":509,"line":2488},116,[507,2490,2060],{"class":572},[507,2492,580],{"class":517},[507,2494,2495],{"class":730},"\"single particle ground state energy: \"",[507,2497,2498],{"class":517},", gs_en)\n",[507,2500,2502,2505],{"class":509,"line":2501},117,[507,2503,2504],{"class":513},"    return",[507,2506,2507],{"class":517}," gs_en\n",[13,2509,2511],{"id":2510},"step-1-map-classical-inputs-to-a-quantum-problem","Step 1: Map classical inputs to a quantum problem",[2513,2514,2516],"h3",{"id":2515},"the-krylov-space","The Krylov space",[18,2518,2519,2520,2613,2614,2643,2644,2673,2674,2733,2734,53],{},"The Krylov space ",[507,2521,2524,2553],{"className":2522},[2523],"katex",[507,2525,2528],{"className":2526},[2527],"katex-mathml",[2529,2530,2532],"math",{"xmlns":2531},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[2533,2534,2535,2548],"semantics",{},[2536,2537,2538],"mrow",{},[2539,2540,2541,2546],"msup",{},[2542,2543,2545],"mi",{"mathvariant":2544},"script","K",[2542,2547,2216],{},[2549,2550,2552],"annotation",{"encoding":2551},"application\u002Fx-tex","\\mathcal{K}^r",[507,2554,2558],{"className":2555,"ariaHidden":2557},[2556],"katex-html","true",[507,2559,2562,2567],{"className":2560},[2561],"base",[507,2563],{"className":2564,"style":2566},[2565],"strut","height:0.6833em;",[507,2568,2571,2576],{"className":2569},[2570],"mord",[507,2572,2545],{"className":2573,"style":2575},[2570,2574],"mathcal","margin-right:0.0144em;",[507,2577,2580],{"className":2578},[2579],"msupsub",[507,2581,2584],{"className":2582},[2583],"vlist-t",[507,2585,2588],{"className":2586},[2587],"vlist-r",[507,2589,2593],{"className":2590,"style":2592},[2591],"vlist","height:0.6644em;",[507,2594,2596,2601],{"style":2595},"top:-3.063em;margin-right:0.05em;",[507,2597],{"className":2598,"style":2600},[2599],"pstrut","height:2.7em;",[507,2602,2608],{"className":2603},[2604,2605,2606,2607],"sizing","reset-size6","size3","mtight",[507,2609,2216],{"className":2610,"style":2612},[2570,2611,2607],"mathnormal","margin-right:0.0278em;"," of order ",[507,2615,2617,2630],{"className":2616},[2523],[507,2618,2620],{"className":2619},[2527],[2529,2621,2622],{"xmlns":2531},[2533,2623,2624,2628],{},[2536,2625,2626],{},[2542,2627,2216],{},[2549,2629,2216],{"encoding":2551},[507,2631,2633],{"className":2632,"ariaHidden":2557},[2556],[507,2634,2636,2640],{"className":2635},[2561],[507,2637],{"className":2638,"style":2639},[2565],"height:0.4306em;",[507,2641,2216],{"className":2642,"style":2612},[2570,2611]," is the space spanned by vectors obtained by multiplying higher powers of a matrix ",[507,2645,2647,2661],{"className":2646},[2523],[507,2648,2650],{"className":2649},[2527],[2529,2651,2652],{"xmlns":2531},[2533,2653,2654,2659],{},[2536,2655,2656],{},[2542,2657,2658],{},"A",[2549,2660,2658],{"encoding":2551},[507,2662,2664],{"className":2663,"ariaHidden":2557},[2556],[507,2665,2667,2670],{"className":2666},[2561],[507,2668],{"className":2669,"style":2566},[2565],[507,2671,2658],{"className":2672},[2570,2611],", up to ",[507,2675,2677,2698],{"className":2676},[2523],[507,2678,2680],{"className":2679},[2527],[2529,2681,2682],{"xmlns":2531},[2533,2683,2684,2695],{},[2536,2685,2686,2688,2692],{},[2542,2687,2216],{},[2689,2690,2691],"mo",{},"−",[2693,2694,625],"mn",{},[2549,2696,2697],{"encoding":2551},"r-1",[507,2699,2701,2723],{"className":2700,"ariaHidden":2557},[2556],[507,2702,2704,2708,2711,2716,2720],{"className":2703},[2561],[507,2705],{"className":2706,"style":2707},[2565],"height:0.6667em;vertical-align:-0.0833em;",[507,2709,2216],{"className":2710,"style":2612},[2570,2611],[507,2712],{"className":2713,"style":2715},[2714],"mspace","margin-right:0.2222em;",[507,2717,2691],{"className":2718},[2719],"mbin",[507,2721],{"className":2722,"style":2715},[2714],[507,2724,2726,2730],{"className":2725},[2561],[507,2727],{"className":2728,"style":2729},[2565],"height:0.6444em;",[507,2731,625],{"className":2732},[2570],", with a reference vector ",[507,2735,2737,2760],{"className":2736},[2523],[507,2738,2740],{"className":2739},[2527],[2529,2741,2742],{"xmlns":2531},[2533,2743,2744,2757],{},[2536,2745,2746,2750,2753],{},[2542,2747,2749],{"mathvariant":2748},"normal","∣",[2542,2751,2752],{},"v",[2689,2754,2756],{"stretchy":2755},"false","⟩",[2549,2758,2759],{"encoding":2551},"\\vert v \\rangle",[507,2761,2763],{"className":2762,"ariaHidden":2557},[2556],[507,2764,2766,2770,2773,2777],{"className":2765},[2561],[507,2767],{"className":2768,"style":2769},[2565],"height:1em;vertical-align:-0.25em;",[507,2771,2749],{"className":2772},[2570],[507,2774,2752],{"className":2775,"style":2776},[2570,2611],"margin-right:0.0359em;",[507,2778,2756],{"className":2779},[2780],"mclose",[507,2782,2785],{"className":2783},[2784],"katex-display",[507,2786,2788,2876],{"className":2787},[2523],[507,2789,2791],{"className":2790},[2527],[2529,2792,2794],{"xmlns":2531,"display":2793},"block",[2533,2795,2796,2873],{},[2536,2797,2798,2804,2806],{},[2539,2799,2800,2802],{},[2542,2801,2545],{"mathvariant":2544},[2542,2803,2216],{},[2689,2805,573],{},[2536,2807,2808,2811,2813,2815,2817,2820,2822,2824,2826,2828,2830,2836,2838,2840,2842,2844,2846,2848,2850,2852,2864,2866,2868,2870],{},[2689,2809,2810],{"fence":2557},"{",[2542,2812,2749],{"mathvariant":2748},[2542,2814,2752],{},[2689,2816,2756],{"stretchy":2755},[2689,2818,2819],{"separator":2557},",",[2542,2821,2658],{},[2542,2823,2749],{"mathvariant":2748},[2542,2825,2752],{},[2689,2827,2756],{"stretchy":2755},[2689,2829,2819],{"separator":2557},[2539,2831,2832,2834],{},[2542,2833,2658],{},[2693,2835,584],{},[2542,2837,2749],{"mathvariant":2748},[2542,2839,2752],{},[2689,2841,2756],{"stretchy":2755},[2689,2843,2819],{"separator":2557},[2542,2845,53],{"mathvariant":2748},[2542,2847,53],{"mathvariant":2748},[2542,2849,53],{"mathvariant":2748},[2689,2851,2819],{"separator":2557},[2539,2853,2854,2856],{},[2542,2855,2658],{},[2536,2857,2858,2860,2862],{},[2542,2859,2216],{},[2689,2861,2691],{},[2693,2863,625],{},[2542,2865,2749],{"mathvariant":2748},[2542,2867,2752],{},[2689,2869,2756],{"stretchy":2755},[2689,2871,2872],{"fence":2557},"}",[2549,2874,2875],{"encoding":2551},"\\mathcal{K}^r = \\left\\{ \\vert v \\rangle, A \\vert v \\rangle, A^2 \\vert v \\rangle, ..., A^{r-1} \\vert v \\rangle \\right\\}",[507,2877,2879,2927],{"className":2878,"ariaHidden":2557},[2556],[507,2880,2882,2886,2916,2920,2924],{"className":2881},[2561],[507,2883],{"className":2884,"style":2885},[2565],"height:0.7144em;",[507,2887,2889,2892],{"className":2888},[2570],[507,2890,2545],{"className":2891,"style":2575},[2570,2574],[507,2893,2895],{"className":2894},[2579],[507,2896,2898],{"className":2897},[2583],[507,2899,2901],{"className":2900},[2587],[507,2902,2904],{"className":2903,"style":2885},[2591],[507,2905,2907,2910],{"style":2906},"top:-3.113em;margin-right:0.05em;",[507,2908],{"className":2909,"style":2600},[2599],[507,2911,2913],{"className":2912},[2604,2605,2606,2607],[507,2914,2216],{"className":2915,"style":2612},[2570,2611,2607],[507,2917],{"className":2918,"style":2919},[2714],"margin-right:0.2778em;",[507,2921,573],{"className":2922},[2923],"mrel",[507,2925],{"className":2926,"style":2919},[2714],[507,2928,2930,2934],{"className":2929},[2561],[507,2931],{"className":2932,"style":2933},[2565],"height:1.2141em;vertical-align:-0.35em;",[507,2935,2938,2949,2952,2955,2958,2962,2966,2969,2972,2975,2978,2981,2984,3014,3017,3020,3023,3026,3029,3033,3036,3039,3077,3080,3083,3086],{"className":2936},[2937],"minner",[507,2939,2944],{"className":2940,"style":2943},[2941,2942],"mopen","delimcenter","top:0em;",[507,2945,2810],{"className":2946},[2947,2948],"delimsizing","size1",[507,2950,2749],{"className":2951},[2570],[507,2953,2752],{"className":2954,"style":2776},[2570,2611],[507,2956,2756],{"className":2957},[2780],[507,2959,2819],{"className":2960},[2961],"mpunct",[507,2963],{"className":2964,"style":2965},[2714],"margin-right:0.1667em;",[507,2967,2658],{"className":2968},[2570,2611],[507,2970,2749],{"className":2971},[2570],[507,2973,2752],{"className":2974,"style":2776},[2570,2611],[507,2976,2756],{"className":2977},[2780],[507,2979,2819],{"className":2980},[2961],[507,2982],{"className":2983,"style":2965},[2714],[507,2985,2987,2990],{"className":2986},[2570],[507,2988,2658],{"className":2989},[2570,2611],[507,2991,2993],{"className":2992},[2579],[507,2994,2996],{"className":2995},[2583],[507,2997,2999],{"className":2998},[2587],[507,3000,3003],{"className":3001,"style":3002},[2591],"height:0.8641em;",[507,3004,3005,3008],{"style":2906},[507,3006],{"className":3007,"style":2600},[2599],[507,3009,3011],{"className":3010},[2604,2605,2606,2607],[507,3012,584],{"className":3013},[2570,2607],[507,3015,2749],{"className":3016},[2570],[507,3018,2752],{"className":3019,"style":2776},[2570,2611],[507,3021,2756],{"className":3022},[2780],[507,3024,2819],{"className":3025},[2961],[507,3027],{"className":3028,"style":2965},[2714],[507,3030,3032],{"className":3031},[2570],"...",[507,3034,2819],{"className":3035},[2961],[507,3037],{"className":3038,"style":2965},[2714],[507,3040,3042,3045],{"className":3041},[2570],[507,3043,2658],{"className":3044},[2570,2611],[507,3046,3048],{"className":3047},[2579],[507,3049,3051],{"className":3050},[2583],[507,3052,3054],{"className":3053},[2587],[507,3055,3057],{"className":3056,"style":3002},[2591],[507,3058,3059,3062],{"style":2906},[507,3060],{"className":3061,"style":2600},[2599],[507,3063,3065],{"className":3064},[2604,2605,2606,2607],[507,3066,3068,3071,3074],{"className":3067},[2570,2607],[507,3069,2216],{"className":3070,"style":2612},[2570,2611,2607],[507,3072,2691],{"className":3073},[2719,2607],[507,3075,625],{"className":3076},[2570,2607],[507,3078,2749],{"className":3079},[2570],[507,3081,2752],{"className":3082,"style":2776},[2570,2611],[507,3084,2756],{"className":3085},[2780],[507,3087,3089],{"className":3088,"style":2943},[2780,2942],[507,3090,2872],{"className":3091},[2947,2948],[18,3093,3094,3095,3123,3124,3154,3155,3235,3236,3264,3265,3372,3373,3443,3444,3472,3473,3574,3575,3578,3579,3607,3608,53],{},"If the matrix ",[507,3096,3098,3111],{"className":3097},[2523],[507,3099,3101],{"className":3100},[2527],[2529,3102,3103],{"xmlns":2531},[2533,3104,3105,3109],{},[2536,3106,3107],{},[2542,3108,2658],{},[2549,3110,2658],{"encoding":2551},[507,3112,3114],{"className":3113,"ariaHidden":2557},[2556],[507,3115,3117,3120],{"className":3116},[2561],[507,3118],{"className":3119,"style":2566},[2565],[507,3121,2658],{"className":3122},[2570,2611]," is the Hamiltonian ",[507,3125,3127,3141],{"className":3126},[2523],[507,3128,3130],{"className":3129},[2527],[2529,3131,3132],{"xmlns":2531},[2533,3133,3134,3139],{},[2536,3135,3136],{},[2542,3137,3138],{},"H",[2549,3140,3138],{"encoding":2551},[507,3142,3144],{"className":3143,"ariaHidden":2557},[2556],[507,3145,3147,3150],{"className":3146},[2561],[507,3148],{"className":3149,"style":2566},[2565],[507,3151,3138],{"className":3152,"style":3153},[2570,2611],"margin-right:0.0813em;",", we'll refer to the corresponding space as the power Krylov space ",[507,3156,3158,3178],{"className":3157},[2523],[507,3159,3161],{"className":3160},[2527],[2529,3162,3163],{"xmlns":2531},[2533,3164,3165,3175],{},[2536,3166,3167],{},[3168,3169,3170,3172],"msub",{},[2542,3171,2545],{"mathvariant":2544},[2542,3173,3174],{},"P",[2549,3176,3177],{"encoding":2551},"\\mathcal{K}_P",[507,3179,3181],{"className":3180,"ariaHidden":2557},[2556],[507,3182,3184,3188],{"className":3183},[2561],[507,3185],{"className":3186,"style":3187},[2565],"height:0.8333em;vertical-align:-0.15em;",[507,3189,3191,3194],{"className":3190},[2570],[507,3192,2545],{"className":3193,"style":2575},[2570,2574],[507,3195,3197],{"className":3196},[2579],[507,3198,3201,3226],{"className":3199},[2583,3200],"vlist-t2",[507,3202,3204,3221],{"className":3203},[2587],[507,3205,3208],{"className":3206,"style":3207},[2591],"height:0.3283em;",[507,3209,3211,3214],{"style":3210},"top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;",[507,3212],{"className":3213,"style":2600},[2599],[507,3215,3217],{"className":3216},[2604,2605,2606,2607],[507,3218,3174],{"className":3219,"style":3220},[2570,2611,2607],"margin-right:0.1389em;",[507,3222,3225],{"className":3223},[3224],"vlist-s","​",[507,3227,3229],{"className":3228},[2587],[507,3230,3233],{"className":3231,"style":3232},[2591],"height:0.15em;",[507,3234],{},". In the case where ",[507,3237,3239,3252],{"className":3238},[2523],[507,3240,3242],{"className":3241},[2527],[2529,3243,3244],{"xmlns":2531},[2533,3245,3246,3250],{},[2536,3247,3248],{},[2542,3249,2658],{},[2549,3251,2658],{"encoding":2551},[507,3253,3255],{"className":3254,"ariaHidden":2557},[2556],[507,3256,3258,3261],{"className":3257},[2561],[507,3259],{"className":3260,"style":2566},[2565],[507,3262,2658],{"className":3263},[2570,2611]," is the time-evolution operator generated by the Hamiltonian ",[507,3266,3268,3302],{"className":3267},[2523],[507,3269,3271],{"className":3270},[2527],[2529,3272,3273],{"xmlns":2531},[2533,3274,3275,3299],{},[2536,3276,3277,3280,3282],{},[2542,3278,3279],{},"U",[2689,3281,573],{},[2539,3283,3284,3287],{},[2542,3285,3286],{},"e",[2536,3288,3289,3291,3294,3296],{},[2689,3290,2691],{},[2542,3292,3293],{},"i",[2542,3295,3138],{},[2542,3297,3298],{},"t",[2549,3300,3301],{"encoding":2551},"U=e^{-iHt}",[507,3303,3305,3324],{"className":3304,"ariaHidden":2557},[2556],[507,3306,3308,3311,3315,3318,3321],{"className":3307},[2561],[507,3309],{"className":3310,"style":2566},[2565],[507,3312,3279],{"className":3313,"style":3314},[2570,2611],"margin-right:0.109em;",[507,3316],{"className":3317,"style":2919},[2714],[507,3319,573],{"className":3320},[2923],[507,3322],{"className":3323,"style":2919},[2714],[507,3325,3327,3331],{"className":3326},[2561],[507,3328],{"className":3329,"style":3330},[2565],"height:0.8413em;",[507,3332,3334,3337],{"className":3333},[2570],[507,3335,3286],{"className":3336},[2570,2611],[507,3338,3340],{"className":3339},[2579],[507,3341,3343],{"className":3342},[2583],[507,3344,3346],{"className":3345},[2587],[507,3347,3349],{"className":3348,"style":3330},[2591],[507,3350,3351,3354],{"style":2595},[507,3352],{"className":3353,"style":2600},[2599],[507,3355,3357],{"className":3356},[2604,2605,2606,2607],[507,3358,3360,3363,3366,3369],{"className":3359},[2570,2607],[507,3361,2691],{"className":3362},[2570,2607],[507,3364,3293],{"className":3365},[2570,2611,2607],[507,3367,3138],{"className":3368,"style":3153},[2570,2611,2607],[507,3370,3298],{"className":3371},[2570,2611,2607],", we'll refer to the space as the unitary Krylov space ",[507,3374,3376,3394],{"className":3375},[2523],[507,3377,3379],{"className":3378},[2527],[2529,3380,3381],{"xmlns":2531},[2533,3382,3383,3391],{},[2536,3384,3385],{},[3168,3386,3387,3389],{},[2542,3388,2545],{"mathvariant":2544},[2542,3390,3279],{},[2549,3392,3393],{"encoding":2551},"\\mathcal{K}_U",[507,3395,3397],{"className":3396,"ariaHidden":2557},[2556],[507,3398,3400,3403],{"className":3399},[2561],[507,3401],{"className":3402,"style":3187},[2565],[507,3404,3406,3409],{"className":3405},[2570],[507,3407,2545],{"className":3408,"style":2575},[2570,2574],[507,3410,3412],{"className":3411},[2579],[507,3413,3415,3435],{"className":3414},[2583,3200],[507,3416,3418,3432],{"className":3417},[2587],[507,3419,3421],{"className":3420,"style":3207},[2591],[507,3422,3423,3426],{"style":3210},[507,3424],{"className":3425,"style":2600},[2599],[507,3427,3429],{"className":3428},[2604,2605,2606,2607],[507,3430,3279],{"className":3431,"style":3314},[2570,2611,2607],[507,3433,3225],{"className":3434},[3224],[507,3436,3438],{"className":3437},[2587],[507,3439,3441],{"className":3440,"style":3232},[2591],[507,3442],{},". The power Krylov subspace that we use classically cannot be generated directly on a quantum computer as ",[507,3445,3447,3460],{"className":3446},[2523],[507,3448,3450],{"className":3449},[2527],[2529,3451,3452],{"xmlns":2531},[2533,3453,3454,3458],{},[2536,3455,3456],{},[2542,3457,3138],{},[2549,3459,3138],{"encoding":2551},[507,3461,3463],{"className":3462,"ariaHidden":2557},[2556],[507,3464,3466,3469],{"className":3465},[2561],[507,3467],{"className":3468,"style":2566},[2565],[507,3470,3138],{"className":3471,"style":3153},[2570,2611]," is not a unitary operator. Instead, we can use the time-evolution operator ",[507,3474,3476,3506],{"className":3475},[2523],[507,3477,3479],{"className":3478},[2527],[2529,3480,3481],{"xmlns":2531},[2533,3482,3483,3503],{},[2536,3484,3485,3487,3489],{},[2542,3486,3279],{},[2689,3488,573],{},[2539,3490,3491,3493],{},[2542,3492,3286],{},[2536,3494,3495,3497,3499,3501],{},[2689,3496,2691],{},[2542,3498,3293],{},[2542,3500,3138],{},[2542,3502,3298],{},[2549,3504,3505],{"encoding":2551},"U = e^{-iHt}",[507,3507,3509,3527],{"className":3508,"ariaHidden":2557},[2556],[507,3510,3512,3515,3518,3521,3524],{"className":3511},[2561],[507,3513],{"className":3514,"style":2566},[2565],[507,3516,3279],{"className":3517,"style":3314},[2570,2611],[507,3519],{"className":3520,"style":2919},[2714],[507,3522,573],{"className":3523},[2923],[507,3525],{"className":3526,"style":2919},[2714],[507,3528,3530,3533],{"className":3529},[2561],[507,3531],{"className":3532,"style":3330},[2565],[507,3534,3536,3539],{"className":3535},[2570],[507,3537,3286],{"className":3538},[2570,2611],[507,3540,3542],{"className":3541},[2579],[507,3543,3545],{"className":3544},[2583],[507,3546,3548],{"className":3547},[2587],[507,3549,3551],{"className":3550,"style":3330},[2591],[507,3552,3553,3556],{"style":2595},[507,3554],{"className":3555,"style":2600},[2599],[507,3557,3559],{"className":3558},[2604,2605,2606,2607],[507,3560,3562,3565,3568,3571],{"className":3561},[2570,2607],[507,3563,2691],{"className":3564},[2570,2607],[507,3566,3293],{"className":3567},[2570,2611,2607],[507,3569,3138],{"className":3570,"style":3153},[2570,2611,2607],[507,3572,3298],{"className":3573},[2570,2611,2607]," which can be shown to give similar ",[49,3576,3577],{"href":1091},"convergence guarantees"," as the power method. Powers of ",[507,3580,3582,3595],{"className":3581},[2523],[507,3583,3585],{"className":3584},[2527],[2529,3586,3587],{"xmlns":2531},[2533,3588,3589,3593],{},[2536,3590,3591],{},[2542,3592,3279],{},[2549,3594,3279],{"encoding":2551},[507,3596,3598],{"className":3597,"ariaHidden":2557},[2556],[507,3599,3601,3604],{"className":3600},[2561],[507,3602],{"className":3603,"style":2566},[2565],[507,3605,3279],{"className":3606,"style":3314},[2570,2611]," then become different time steps ",[507,3609,3611,3653],{"className":3610},[2523],[507,3612,3614],{"className":3613},[2527],[2529,3615,3616],{"xmlns":2531},[2533,3617,3618,3650],{},[2536,3619,3620,3627,3629],{},[2539,3621,3622,3624],{},[2542,3623,3279],{},[2542,3625,3626],{},"k",[2689,3628,573],{},[2539,3630,3631,3633],{},[2542,3632,3286],{},[2536,3634,3635,3637,3639,3641,3643,3645,3647],{},[2689,3636,2691],{},[2542,3638,3293],{},[2542,3640,3138],{},[2689,3642,580],{"stretchy":2755},[2542,3644,3626],{},[2542,3646,3298],{},[2689,3648,3649],{"stretchy":2755},")",[2549,3651,3652],{"encoding":2551},"U^k = e^{-iH(kt)}",[507,3654,3656,3702],{"className":3655,"ariaHidden":2557},[2556],[507,3657,3659,3663,3693,3696,3699],{"className":3658},[2561],[507,3660],{"className":3661,"style":3662},[2565],"height:0.8491em;",[507,3664,3666,3669],{"className":3665},[2570],[507,3667,3279],{"className":3668,"style":3314},[2570,2611],[507,3670,3672],{"className":3671},[2579],[507,3673,3675],{"className":3674},[2583],[507,3676,3678],{"className":3677},[2587],[507,3679,3681],{"className":3680,"style":3662},[2591],[507,3682,3683,3686],{"style":2595},[507,3684],{"className":3685,"style":2600},[2599],[507,3687,3689],{"className":3688},[2604,2605,2606,2607],[507,3690,3626],{"className":3691,"style":3692},[2570,2611,2607],"margin-right:0.0315em;",[507,3694],{"className":3695,"style":2919},[2714],[507,3697,573],{"className":3698},[2923],[507,3700],{"className":3701,"style":2919},[2714],[507,3703,3705,3709],{"className":3704},[2561],[507,3706],{"className":3707,"style":3708},[2565],"height:0.888em;",[507,3710,3712,3715],{"className":3711},[2570],[507,3713,3286],{"className":3714},[2570,2611],[507,3716,3718],{"className":3717},[2579],[507,3719,3721],{"className":3720},[2583],[507,3722,3724],{"className":3723},[2587],[507,3725,3727],{"className":3726,"style":3708},[2591],[507,3728,3729,3732],{"style":2595},[507,3730],{"className":3731,"style":2600},[2599],[507,3733,3735],{"className":3734},[2604,2605,2606,2607],[507,3736,3738,3741,3744,3747,3750,3753,3756],{"className":3737},[2570,2607],[507,3739,2691],{"className":3740},[2570,2607],[507,3742,3293],{"className":3743},[2570,2611,2607],[507,3745,3138],{"className":3746,"style":3153},[2570,2611,2607],[507,3748,580],{"className":3749},[2941,2607],[507,3751,3626],{"className":3752,"style":3692},[2570,2611,2607],[507,3754,3298],{"className":3755},[2570,2611,2607],[507,3757,3649],{"className":3758},[2780,2607],[507,3760,3762],{"className":3761},[2784],[507,3763,3765,3853],{"className":3764},[2523],[507,3766,3768],{"className":3767},[2527],[2529,3769,3770],{"xmlns":2531,"display":2793},[2533,3771,3772,3850],{},[2536,3773,3774,3783,3785],{},[3775,3776,3777,3779,3781],"msubsup",{},[2542,3778,2545],{"mathvariant":2544},[2542,3780,3279],{},[2542,3782,2216],{},[2689,3784,573],{},[2536,3786,3787,3789,3791,3794,3796,3798,3800,3802,3804,3806,3808,3814,3816,3818,3820,3822,3824,3826,3828,3830,3842,3844,3846,3848],{},[2689,3788,2810],{"fence":2557},[2542,3790,2749],{"mathvariant":2748},[2542,3792,3793],{},"ψ",[2689,3795,2756],{"stretchy":2755},[2689,3797,2819],{"separator":2557},[2542,3799,3279],{},[2542,3801,2749],{"mathvariant":2748},[2542,3803,3793],{},[2689,3805,2756],{"stretchy":2755},[2689,3807,2819],{"separator":2557},[2539,3809,3810,3812],{},[2542,3811,3279],{},[2693,3813,584],{},[2542,3815,2749],{"mathvariant":2748},[2542,3817,3793],{},[2689,3819,2756],{"stretchy":2755},[2689,3821,2819],{"separator":2557},[2542,3823,53],{"mathvariant":2748},[2542,3825,53],{"mathvariant":2748},[2542,3827,53],{"mathvariant":2748},[2689,3829,2819],{"separator":2557},[2539,3831,3832,3834],{},[2542,3833,3279],{},[2536,3835,3836,3838,3840],{},[2542,3837,2216],{},[2689,3839,2691],{},[2693,3841,625],{},[2542,3843,2749],{"mathvariant":2748},[2542,3845,3793],{},[2689,3847,2756],{"stretchy":2755},[2689,3849,2872],{"fence":2557},[2549,3851,3852],{"encoding":2551},"\\mathcal{K}_U^r = \\left\\{ \\vert \\psi \\rangle, U \\vert \\psi \\rangle, U^2 \\vert \\psi \\rangle, ..., U^{r-1} \\vert \\psi \\rangle \\right\\}",[507,3854,3856,3925],{"className":3855,"ariaHidden":2557},[2556],[507,3857,3859,3863,3916,3919,3922],{"className":3858},[2561],[507,3860],{"className":3861,"style":3862},[2565],"height:0.9614em;vertical-align:-0.247em;",[507,3864,3866,3869],{"className":3865},[2570],[507,3867,2545],{"className":3868,"style":2575},[2570,2574],[507,3870,3872],{"className":3871},[2579],[507,3873,3875,3907],{"className":3874},[2583,3200],[507,3876,3878,3904],{"className":3877},[2587],[507,3879,3881,3893],{"className":3880,"style":2885},[2591],[507,3882,3884,3887],{"style":3883},"top:-2.453em;margin-left:-0.0144em;margin-right:0.05em;",[507,3885],{"className":3886,"style":2600},[2599],[507,3888,3890],{"className":3889},[2604,2605,2606,2607],[507,3891,3279],{"className":3892,"style":3314},[2570,2611,2607],[507,3894,3895,3898],{"style":2906},[507,3896],{"className":3897,"style":2600},[2599],[507,3899,3901],{"className":3900},[2604,2605,2606,2607],[507,3902,2216],{"className":3903,"style":2612},[2570,2611,2607],[507,3905,3225],{"className":3906},[3224],[507,3908,3910],{"className":3909},[2587],[507,3911,3914],{"className":3912,"style":3913},[2591],"height:0.247em;",[507,3915],{},[507,3917],{"className":3918,"style":2919},[2714],[507,3920,573],{"className":3921},[2923],[507,3923],{"className":3924,"style":2919},[2714],[507,3926,3928,3931],{"className":3927},[2561],[507,3929],{"className":3930,"style":2933},[2565],[507,3932,3934,3940,3943,3946,3949,3952,3955,3958,3961,3964,3967,3970,3973,4002,4005,4008,4011,4014,4017,4020,4023,4026,4064,4067,4070,4073],{"className":3933},[2937],[507,3935,3937],{"className":3936,"style":2943},[2941,2942],[507,3938,2810],{"className":3939},[2947,2948],[507,3941,2749],{"className":3942},[2570],[507,3944,3793],{"className":3945,"style":2776},[2570,2611],[507,3947,2756],{"className":3948},[2780],[507,3950,2819],{"className":3951},[2961],[507,3953],{"className":3954,"style":2965},[2714],[507,3956,3279],{"className":3957,"style":3314},[2570,2611],[507,3959,2749],{"className":3960},[2570],[507,3962,3793],{"className":3963,"style":2776},[2570,2611],[507,3965,2756],{"className":3966},[2780],[507,3968,2819],{"className":3969},[2961],[507,3971],{"className":3972,"style":2965},[2714],[507,3974,3976,3979],{"className":3975},[2570],[507,3977,3279],{"className":3978,"style":3314},[2570,2611],[507,3980,3982],{"className":3981},[2579],[507,3983,3985],{"className":3984},[2583],[507,3986,3988],{"className":3987},[2587],[507,3989,3991],{"className":3990,"style":3002},[2591],[507,3992,3993,3996],{"style":2906},[507,3994],{"className":3995,"style":2600},[2599],[507,3997,3999],{"className":3998},[2604,2605,2606,2607],[507,4000,584],{"className":4001},[2570,2607],[507,4003,2749],{"className":4004},[2570],[507,4006,3793],{"className":4007,"style":2776},[2570,2611],[507,4009,2756],{"className":4010},[2780],[507,4012,2819],{"className":4013},[2961],[507,4015],{"className":4016,"style":2965},[2714],[507,4018,3032],{"className":4019},[2570],[507,4021,2819],{"className":4022},[2961],[507,4024],{"className":4025,"style":2965},[2714],[507,4027,4029,4032],{"className":4028},[2570],[507,4030,3279],{"className":4031,"style":3314},[2570,2611],[507,4033,4035],{"className":4034},[2579],[507,4036,4038],{"className":4037},[2583],[507,4039,4041],{"className":4040},[2587],[507,4042,4044],{"className":4043,"style":3002},[2591],[507,4045,4046,4049],{"style":2906},[507,4047],{"className":4048,"style":2600},[2599],[507,4050,4052],{"className":4051},[2604,2605,2606,2607],[507,4053,4055,4058,4061],{"className":4054},[2570,2607],[507,4056,2216],{"className":4057,"style":2612},[2570,2611,2607],[507,4059,2691],{"className":4060},[2719,2607],[507,4062,625],{"className":4063},[2570,2607],[507,4065,2749],{"className":4066},[2570],[507,4068,3793],{"className":4069,"style":2776},[2570,2611],[507,4071,2756],{"className":4072},[2780],[507,4074,4076],{"className":4075,"style":2943},[2780,2942],[507,4077,2872],{"className":4078},[2947,2948],[18,4080,4081],{},"See the Appendix for a detailed derivation of how the unitary Krylov space allows to represents low-energy eigenstates accurately.",[2513,4083,4085],{"id":4084},"krylov-quantum-diagonalization-algorithm","Krylov quantum diagonalization algorithm",[18,4087,4088,4089,4117,4118,4187,4188,4257,4258,4328,4329,4389],{},"Given an Hamiltonian ",[507,4090,4092,4105],{"className":4091},[2523],[507,4093,4095],{"className":4094},[2527],[2529,4096,4097],{"xmlns":2531},[2533,4098,4099,4103],{},[2536,4100,4101],{},[2542,4102,3138],{},[2549,4104,3138],{"encoding":2551},[507,4106,4108],{"className":4107,"ariaHidden":2557},[2556],[507,4109,4111,4114],{"className":4110},[2561],[507,4112],{"className":4113,"style":2566},[2565],[507,4115,3138],{"className":4116,"style":3153},[2570,2611]," that we wish to diagonalize, first we consider the corresponding unitary Krylov space ",[507,4119,4121,4138],{"className":4120},[2523],[507,4122,4124],{"className":4123},[2527],[2529,4125,4126],{"xmlns":2531},[2533,4127,4128,4136],{},[2536,4129,4130],{},[3168,4131,4132,4134],{},[2542,4133,2545],{"mathvariant":2544},[2542,4135,3279],{},[2549,4137,3393],{"encoding":2551},[507,4139,4141],{"className":4140,"ariaHidden":2557},[2556],[507,4142,4144,4147],{"className":4143},[2561],[507,4145],{"className":4146,"style":3187},[2565],[507,4148,4150,4153],{"className":4149},[2570],[507,4151,2545],{"className":4152,"style":2575},[2570,2574],[507,4154,4156],{"className":4155},[2579],[507,4157,4159,4179],{"className":4158},[2583,3200],[507,4160,4162,4176],{"className":4161},[2587],[507,4163,4165],{"className":4164,"style":3207},[2591],[507,4166,4167,4170],{"style":3210},[507,4168],{"className":4169,"style":2600},[2599],[507,4171,4173],{"className":4172},[2604,2605,2606,2607],[507,4174,3279],{"className":4175,"style":3314},[2570,2611,2607],[507,4177,3225],{"className":4178},[3224],[507,4180,4182],{"className":4181},[2587],[507,4183,4185],{"className":4184,"style":3232},[2591],[507,4186],{},". The goal is to find a compact representation of the Hamiltonian in ",[507,4189,4191,4208],{"className":4190},[2523],[507,4192,4194],{"className":4193},[2527],[2529,4195,4196],{"xmlns":2531},[2533,4197,4198,4206],{},[2536,4199,4200],{},[3168,4201,4202,4204],{},[2542,4203,2545],{"mathvariant":2544},[2542,4205,3279],{},[2549,4207,3393],{"encoding":2551},[507,4209,4211],{"className":4210,"ariaHidden":2557},[2556],[507,4212,4214,4217],{"className":4213},[2561],[507,4215],{"className":4216,"style":3187},[2565],[507,4218,4220,4223],{"className":4219},[2570],[507,4221,2545],{"className":4222,"style":2575},[2570,2574],[507,4224,4226],{"className":4225},[2579],[507,4227,4229,4249],{"className":4228},[2583,3200],[507,4230,4232,4246],{"className":4231},[2587],[507,4233,4235],{"className":4234,"style":3207},[2591],[507,4236,4237,4240],{"style":3210},[507,4238],{"className":4239,"style":2600},[2599],[507,4241,4243],{"className":4242},[2604,2605,2606,2607],[507,4244,3279],{"className":4245,"style":3314},[2570,2611,2607],[507,4247,3225],{"className":4248},[3224],[507,4250,4252],{"className":4251},[2587],[507,4253,4255],{"className":4254,"style":3232},[2591],[507,4256],{},", which we'll refer to as ",[507,4259,4261,4281],{"className":4260},[2523],[507,4262,4264],{"className":4263},[2527],[2529,4265,4266],{"xmlns":2531},[2533,4267,4268,4278],{},[2536,4269,4270],{},[4271,4272,4273,4275],"mover",{"accent":2557},[2542,4274,3138],{},[2689,4276,4277],{},"~",[2549,4279,4280],{"encoding":2551},"\\tilde{H}",[507,4282,4284],{"className":4283,"ariaHidden":2557},[2556],[507,4285,4287,4291],{"className":4286},[2561],[507,4288],{"className":4289,"style":4290},[2565],"height:0.9202em;",[507,4292,4295],{"className":4293},[2570,4294],"accent",[507,4296,4298],{"className":4297},[2583],[507,4299,4301],{"className":4300},[2587],[507,4302,4304,4314],{"className":4303,"style":4290},[2591],[507,4305,4307,4311],{"style":4306},"top:-3em;",[507,4308],{"className":4309,"style":4310},[2599],"height:3em;",[507,4312,3138],{"className":4313,"style":3153},[2570,2611],[507,4315,4317,4320],{"style":4316},"top:-3.6023em;",[507,4318],{"className":4319,"style":4310},[2599],[507,4321,4325],{"className":4322,"style":4324},[4323],"accent-body","left:-0.1944em;",[507,4326,4277],{"className":4327},[2570],". The matrix elements of ",[507,4330,4332,4349],{"className":4331},[2523],[507,4333,4335],{"className":4334},[2527],[2529,4336,4337],{"xmlns":2531},[2533,4338,4339,4347],{},[2536,4340,4341],{},[4271,4342,4343,4345],{"accent":2557},[2542,4344,3138],{},[2689,4346,4277],{},[2549,4348,4280],{"encoding":2551},[507,4350,4352],{"className":4351,"ariaHidden":2557},[2556],[507,4353,4355,4358],{"className":4354},[2561],[507,4356],{"className":4357,"style":4290},[2565],[507,4359,4361],{"className":4360},[2570,4294],[507,4362,4364],{"className":4363},[2583],[507,4365,4367],{"className":4366},[2587],[507,4368,4370,4378],{"className":4369,"style":4290},[2591],[507,4371,4372,4375],{"style":4306},[507,4373],{"className":4374,"style":4310},[2599],[507,4376,3138],{"className":4377,"style":3153},[2570,2611],[507,4379,4380,4383],{"style":4316},[507,4381],{"className":4382,"style":4310},[2599],[507,4384,4386],{"className":4385,"style":4324},[4323],[507,4387,4277],{"className":4388},[2570],", the projection of the Hamiltonian in the Krylov space, can be calculated by calculating the following expectation values",[507,4391,4393],{"className":4392},[2784],[507,4394,4396,4451],{"className":4395},[2523],[507,4397,4399],{"className":4398},[2527],[2529,4400,4401],{"xmlns":2531,"display":2793},[2533,4402,4403,4448],{},[2536,4404,4405,4421,4423,4426,4432,4434,4436,4438,4444,4446],{},[3168,4406,4407,4413],{},[4271,4408,4409,4411],{"accent":2557},[2542,4410,3138],{},[2689,4412,4277],{},[2536,4414,4415,4418],{},[2542,4416,4417],{},"m",[2542,4419,4420],{},"n",[2689,4422,573],{},[2689,4424,4425],{"stretchy":2755},"⟨",[3168,4427,4428,4430],{},[2542,4429,3793],{},[2542,4431,4417],{},[2542,4433,2749],{"mathvariant":2748},[2542,4435,3138],{},[2542,4437,2749],{"mathvariant":2748},[3168,4439,4440,4442],{},[2542,4441,3793],{},[2542,4443,4420],{},[2689,4445,2756],{"stretchy":2755},[2689,4447,573],{},[2549,4449,4450],{"encoding":2551},"\\tilde{H}_{mn} = \\langle \\psi_m \\vert H \\vert \\psi_n \\rangle =",[507,4452,4454,4543],{"className":4453,"ariaHidden":2557},[2556],[507,4455,4457,4461,4534,4537,4540],{"className":4456},[2561],[507,4458],{"className":4459,"style":4460},[2565],"height:1.0702em;vertical-align:-0.15em;",[507,4462,4464,4495],{"className":4463},[2570],[507,4465,4467],{"className":4466},[2570,4294],[507,4468,4470],{"className":4469},[2583],[507,4471,4473],{"className":4472},[2587],[507,4474,4476,4484],{"className":4475,"style":4290},[2591],[507,4477,4478,4481],{"style":4306},[507,4479],{"className":4480,"style":4310},[2599],[507,4482,3138],{"className":4483,"style":3153},[2570,2611],[507,4485,4486,4489],{"style":4316},[507,4487],{"className":4488,"style":4310},[2599],[507,4490,4492],{"className":4491,"style":4324},[4323],[507,4493,4277],{"className":4494},[2570],[507,4496,4498],{"className":4497},[2579],[507,4499,4501,4526],{"className":4500},[2583,3200],[507,4502,4504,4523],{"className":4503},[2587],[507,4505,4508],{"className":4506,"style":4507},[2591],"height:0.1514em;",[507,4509,4511,4514],{"style":4510},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[507,4512],{"className":4513,"style":2600},[2599],[507,4515,4517],{"className":4516},[2604,2605,2606,2607],[507,4518,4520],{"className":4519},[2570,2607],[507,4521,2693],{"className":4522},[2570,2611,2607],[507,4524,3225],{"className":4525},[3224],[507,4527,4529],{"className":4528},[2587],[507,4530,4532],{"className":4531,"style":3232},[2591],[507,4533],{},[507,4535],{"className":4536,"style":2919},[2714],[507,4538,573],{"className":4539},[2923],[507,4541],{"className":4542,"style":2919},[2714],[507,4544,4546,4549,4552,4593,4596,4599,4602,4642,4645,4648],{"className":4545},[2561],[507,4547],{"className":4548,"style":2769},[2565],[507,4550,4425],{"className":4551},[2941],[507,4553,4555,4558],{"className":4554},[2570],[507,4556,3793],{"className":4557,"style":2776},[2570,2611],[507,4559,4561],{"className":4560},[2579],[507,4562,4564,4585],{"className":4563},[2583,3200],[507,4565,4567,4582],{"className":4566},[2587],[507,4568,4570],{"className":4569,"style":4507},[2591],[507,4571,4573,4576],{"style":4572},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[507,4574],{"className":4575,"style":2600},[2599],[507,4577,4579],{"className":4578},[2604,2605,2606,2607],[507,4580,4417],{"className":4581},[2570,2611,2607],[507,4583,3225],{"className":4584},[3224],[507,4586,4588],{"className":4587},[2587],[507,4589,4591],{"className":4590,"style":3232},[2591],[507,4592],{},[507,4594,2749],{"className":4595},[2570],[507,4597,3138],{"className":4598,"style":3153},[2570,2611],[507,4600,2749],{"className":4601},[2570],[507,4603,4605,4608],{"className":4604},[2570],[507,4606,3793],{"className":4607,"style":2776},[2570,2611],[507,4609,4611],{"className":4610},[2579],[507,4612,4614,4634],{"className":4613},[2583,3200],[507,4615,4617,4631],{"className":4616},[2587],[507,4618,4620],{"className":4619,"style":4507},[2591],[507,4621,4622,4625],{"style":4572},[507,4623],{"className":4624,"style":2600},[2599],[507,4626,4628],{"className":4627},[2604,2605,2606,2607],[507,4629,4420],{"className":4630},[2570,2611,2607],[507,4632,3225],{"className":4633},[3224],[507,4635,4637],{"className":4636},[2587],[507,4638,4640],{"className":4639,"style":3232},[2591],[507,4641],{},[507,4643,2756],{"className":4644},[2780],[507,4646],{"className":4647,"style":2919},[2714],[507,4649,573],{"className":4650},[2923],[507,4652,4654],{"className":4653},[2784],[507,4655,4657,4719],{"className":4656},[2523],[507,4658,4660],{"className":4659},[2527],[2529,4661,4662],{"xmlns":2531,"display":2793},[2533,4663,4664,4716],{},[2536,4665,4666,4668,4670,4672,4674,4690,4692,4710,4712,4714],{},[2689,4667,573],{},[2689,4669,4425],{"stretchy":2755},[2542,4671,3793],{},[2542,4673,2749],{"mathvariant":2748},[2539,4675,4676,4678],{},[2542,4677,3286],{},[2536,4679,4680,4682,4684],{},[2542,4681,3293],{},[2542,4683,3138],{},[3168,4685,4686,4688],{},[2542,4687,3298],{},[2542,4689,4417],{},[2542,4691,3138],{},[2539,4693,4694,4696],{},[2542,4695,3286],{},[2536,4697,4698,4700,4702,4704],{},[2689,4699,2691],{},[2542,4701,3293],{},[2542,4703,3138],{},[3168,4705,4706,4708],{},[2542,4707,3298],{},[2542,4709,4420],{},[2542,4711,2749],{"mathvariant":2748},[2542,4713,3793],{},[2689,4715,2756],{"stretchy":2755},[2549,4717,4718],{"encoding":2551},"= \\langle \\psi \\vert  e^{i H t_m}   H e^{-i H t_n} \\vert \\psi \\rangle",[507,4720,4722,4735],{"className":4721,"ariaHidden":2557},[2556],[507,4723,4725,4729,4732],{"className":4724},[2561],[507,4726],{"className":4727,"style":4728},[2565],"height:0.3669em;",[507,4730,573],{"className":4731},[2923],[507,4733],{"className":4734,"style":2919},[2714],[507,4736,4738,4742,4745,4748,4751,4832,4835,4913,4916,4919],{"className":4737},[2561],[507,4739],{"className":4740,"style":4741},[2565],"height:1.1413em;vertical-align:-0.25em;",[507,4743,4425],{"className":4744},[2941],[507,4746,3793],{"className":4747,"style":2776},[2570,2611],[507,4749,2749],{"className":4750},[2570],[507,4752,4754,4757],{"className":4753},[2570],[507,4755,3286],{"className":4756},[2570,2611],[507,4758,4760],{"className":4759},[2579],[507,4761,4763],{"className":4762},[2583],[507,4764,4766],{"className":4765},[2587],[507,4767,4770],{"className":4768,"style":4769},[2591],"height:0.8913em;",[507,4771,4772,4775],{"style":2906},[507,4773],{"className":4774,"style":2600},[2599],[507,4776,4778],{"className":4777},[2604,2605,2606,2607],[507,4779,4781,4784,4787],{"className":4780},[2570,2607],[507,4782,3293],{"className":4783},[2570,2611,2607],[507,4785,3138],{"className":4786,"style":3153},[2570,2611,2607],[507,4788,4790,4793],{"className":4789},[2570,2607],[507,4791,3298],{"className":4792},[2570,2611,2607],[507,4794,4796],{"className":4795},[2579],[507,4797,4799,4823],{"className":4798},[2583,3200],[507,4800,4802,4820],{"className":4801},[2587],[507,4803,4806],{"className":4804,"style":4805},[2591],"height:0.1645em;",[507,4807,4809,4813],{"style":4808},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[507,4810],{"className":4811,"style":4812},[2599],"height:2.5em;",[507,4814,4817],{"className":4815},[2604,4816,2948,2607],"reset-size3",[507,4818,4417],{"className":4819},[2570,2611,2607],[507,4821,3225],{"className":4822},[3224],[507,4824,4826],{"className":4825},[2587],[507,4827,4830],{"className":4828,"style":4829},[2591],"height:0.143em;",[507,4831],{},[507,4833,3138],{"className":4834,"style":3153},[2570,2611],[507,4836,4838,4841],{"className":4837},[2570],[507,4839,3286],{"className":4840},[2570,2611],[507,4842,4844],{"className":4843},[2579],[507,4845,4847],{"className":4846},[2583],[507,4848,4850],{"className":4849},[2587],[507,4851,4853],{"className":4852,"style":4769},[2591],[507,4854,4855,4858],{"style":2906},[507,4856],{"className":4857,"style":2600},[2599],[507,4859,4861],{"className":4860},[2604,2605,2606,2607],[507,4862,4864,4867,4870,4873],{"className":4863},[2570,2607],[507,4865,2691],{"className":4866},[2570,2607],[507,4868,3293],{"className":4869},[2570,2611,2607],[507,4871,3138],{"className":4872,"style":3153},[2570,2611,2607],[507,4874,4876,4879],{"className":4875},[2570,2607],[507,4877,3298],{"className":4878},[2570,2611,2607],[507,4880,4882],{"className":4881},[2579],[507,4883,4885,4905],{"className":4884},[2583,3200],[507,4886,4888,4902],{"className":4887},[2587],[507,4889,4891],{"className":4890,"style":4805},[2591],[507,4892,4893,4896],{"style":4808},[507,4894],{"className":4895,"style":4812},[2599],[507,4897,4899],{"className":4898},[2604,4816,2948,2607],[507,4900,4420],{"className":4901},[2570,2611,2607],[507,4903,3225],{"className":4904},[3224],[507,4906,4908],{"className":4907},[2587],[507,4909,4911],{"className":4910,"style":4829},[2591],[507,4912],{},[507,4914,2749],{"className":4915},[2570],[507,4917,3793],{"className":4918,"style":2776},[2570,2611],[507,4920,2756],{"className":4921},[2780],[507,4923,4925],{"className":4924},[2784],[507,4926,4928,4991],{"className":4927},[2523],[507,4929,4931],{"className":4930},[2527],[2529,4932,4933],{"xmlns":2531,"display":2793},[2533,4934,4935,4988],{},[2536,4936,4937,4939,4941,4943,4945,4962,4964,4982,4984,4986],{},[2689,4938,573],{},[2689,4940,4425],{"stretchy":2755},[2542,4942,3793],{},[2542,4944,2749],{"mathvariant":2748},[2539,4946,4947,4949],{},[2542,4948,3286],{},[2536,4950,4951,4953,4955,4957,4960],{},[2542,4952,3293],{},[2542,4954,3138],{},[2542,4956,4417],{},[2542,4958,4959],{},"d",[2542,4961,3298],{},[2542,4963,3138],{},[2539,4965,4966,4968],{},[2542,4967,3286],{},[2536,4969,4970,4972,4974,4976,4978,4980],{},[2689,4971,2691],{},[2542,4973,3293],{},[2542,4975,3138],{},[2542,4977,4420],{},[2542,4979,4959],{},[2542,4981,3298],{},[2542,4983,2749],{"mathvariant":2748},[2542,4985,3793],{},[2689,4987,2756],{"stretchy":2755},[2549,4989,4990],{"encoding":2551},"= \\langle \\psi \\vert  e^{i H m dt}   H e^{-i H n dt} \\vert \\psi \\rangle",[507,4992,4994,5006],{"className":4993,"ariaHidden":2557},[2556],[507,4995,4997,5000,5003],{"className":4996},[2561],[507,4998],{"className":4999,"style":4728},[2565],[507,5001,573],{"className":5002},[2923],[507,5004],{"className":5005,"style":2919},[2714],[507,5007,5009,5013,5016,5019,5022,5067,5070,5117,5120,5123],{"className":5008},[2561],[507,5010],{"className":5011,"style":5012},[2565],"height:1.1491em;vertical-align:-0.25em;",[507,5014,4425],{"className":5015},[2941],[507,5017,3793],{"className":5018,"style":2776},[2570,2611],[507,5020,2749],{"className":5021},[2570],[507,5023,5025,5028],{"className":5024},[2570],[507,5026,3286],{"className":5027},[2570,2611],[507,5029,5031],{"className":5030},[2579],[507,5032,5034],{"className":5033},[2583],[507,5035,5037],{"className":5036},[2587],[507,5038,5041],{"className":5039,"style":5040},[2591],"height:0.8991em;",[507,5042,5043,5046],{"style":2906},[507,5044],{"className":5045,"style":2600},[2599],[507,5047,5049],{"className":5048},[2604,2605,2606,2607],[507,5050,5052,5055,5058,5061,5064],{"className":5051},[2570,2607],[507,5053,3293],{"className":5054},[2570,2611,2607],[507,5056,3138],{"className":5057,"style":3153},[2570,2611,2607],[507,5059,4417],{"className":5060},[2570,2611,2607],[507,5062,4959],{"className":5063},[2570,2611,2607],[507,5065,3298],{"className":5066},[2570,2611,2607],[507,5068,3138],{"className":5069,"style":3153},[2570,2611],[507,5071,5073,5076],{"className":5072},[2570],[507,5074,3286],{"className":5075},[2570,2611],[507,5077,5079],{"className":5078},[2579],[507,5080,5082],{"className":5081},[2583],[507,5083,5085],{"className":5084},[2587],[507,5086,5088],{"className":5087,"style":5040},[2591],[507,5089,5090,5093],{"style":2906},[507,5091],{"className":5092,"style":2600},[2599],[507,5094,5096],{"className":5095},[2604,2605,2606,2607],[507,5097,5099,5102,5105,5108,5111,5114],{"className":5098},[2570,2607],[507,5100,2691],{"className":5101},[2570,2607],[507,5103,3293],{"className":5104},[2570,2611,2607],[507,5106,3138],{"className":5107,"style":3153},[2570,2611,2607],[507,5109,4420],{"className":5110},[2570,2611,2607],[507,5112,4959],{"className":5113},[2570,2611,2607],[507,5115,3298],{"className":5116},[2570,2611,2607],[507,5118,2749],{"className":5119},[2570],[507,5121,3793],{"className":5122,"style":2776},[2570,2611],[507,5124,2756],{"className":5125},[2780],[18,5127,5128,5129,5338,5339,5444,5445,5479,5480,5549,5550,5578,5579,5631,5632,5660,5661,5718,5719,5779,5780,5840,5841],{},"Where ",[507,5130,5132,5180],{"className":5131},[2523],[507,5133,5135],{"className":5134},[2527],[2529,5136,5137],{"xmlns":2531},[2533,5138,5139,5177],{},[2536,5140,5141,5143,5149,5151,5153,5171,5173,5175],{},[2542,5142,2749],{"mathvariant":2748},[3168,5144,5145,5147],{},[2542,5146,3793],{},[2542,5148,4420],{},[2689,5150,2756],{"stretchy":2755},[2689,5152,573],{},[2539,5154,5155,5157],{},[2542,5156,3286],{},[2536,5158,5159,5161,5163,5165],{},[2689,5160,2691],{},[2542,5162,3293],{},[2542,5164,3138],{},[3168,5166,5167,5169],{},[2542,5168,3298],{},[2542,5170,4420],{},[2542,5172,2749],{"mathvariant":2748},[2542,5174,3793],{},[2689,5176,2756],{"stretchy":2755},[2549,5178,5179],{"encoding":2551},"\\vert \\psi_n \\rangle = e^{-i H t_n} \\vert \\psi \\rangle",[507,5181,5183,5244],{"className":5182,"ariaHidden":2557},[2556],[507,5184,5186,5189,5192,5232,5235,5238,5241],{"className":5185},[2561],[507,5187],{"className":5188,"style":2769},[2565],[507,5190,2749],{"className":5191},[2570],[507,5193,5195,5198],{"className":5194},[2570],[507,5196,3793],{"className":5197,"style":2776},[2570,2611],[507,5199,5201],{"className":5200},[2579],[507,5202,5204,5224],{"className":5203},[2583,3200],[507,5205,5207,5221],{"className":5206},[2587],[507,5208,5210],{"className":5209,"style":4507},[2591],[507,5211,5212,5215],{"style":4572},[507,5213],{"className":5214,"style":2600},[2599],[507,5216,5218],{"className":5217},[2604,2605,2606,2607],[507,5219,4420],{"className":5220},[2570,2611,2607],[507,5222,3225],{"className":5223},[3224],[507,5225,5227],{"className":5226},[2587],[507,5228,5230],{"className":5229,"style":3232},[2591],[507,5231],{},[507,5233,2756],{"className":5234},[2780],[507,5236],{"className":5237,"style":2919},[2714],[507,5239,573],{"className":5240},[2923],[507,5242],{"className":5243,"style":2919},[2714],[507,5245,5247,5251,5329,5332,5335],{"className":5246},[2561],[507,5248],{"className":5249,"style":5250},[2565],"height:1.0913em;vertical-align:-0.25em;",[507,5252,5254,5257],{"className":5253},[2570],[507,5255,3286],{"className":5256},[2570,2611],[507,5258,5260],{"className":5259},[2579],[507,5261,5263],{"className":5262},[2583],[507,5264,5266],{"className":5265},[2587],[507,5267,5269],{"className":5268,"style":3330},[2591],[507,5270,5271,5274],{"style":2595},[507,5272],{"className":5273,"style":2600},[2599],[507,5275,5277],{"className":5276},[2604,2605,2606,2607],[507,5278,5280,5283,5286,5289],{"className":5279},[2570,2607],[507,5281,2691],{"className":5282},[2570,2607],[507,5284,3293],{"className":5285},[2570,2611,2607],[507,5287,3138],{"className":5288,"style":3153},[2570,2611,2607],[507,5290,5292,5295],{"className":5291},[2570,2607],[507,5293,3298],{"className":5294},[2570,2611,2607],[507,5296,5298],{"className":5297},[2579],[507,5299,5301,5321],{"className":5300},[2583,3200],[507,5302,5304,5318],{"className":5303},[2587],[507,5305,5307],{"className":5306,"style":4805},[2591],[507,5308,5309,5312],{"style":4808},[507,5310],{"className":5311,"style":4812},[2599],[507,5313,5315],{"className":5314},[2604,4816,2948,2607],[507,5316,4420],{"className":5317},[2570,2611,2607],[507,5319,3225],{"className":5320},[3224],[507,5322,5324],{"className":5323},[2587],[507,5325,5327],{"className":5326,"style":4829},[2591],[507,5328],{},[507,5330,2749],{"className":5331},[2570],[507,5333,3793],{"className":5334,"style":2776},[2570,2611],[507,5336,2756],{"className":5337},[2780]," are the vectors of the unitary Krylov space and ",[507,5340,5342,5368],{"className":5341},[2523],[507,5343,5345],{"className":5344},[2527],[2529,5346,5347],{"xmlns":2531},[2533,5348,5349,5365],{},[2536,5350,5351,5357,5359,5361,5363],{},[3168,5352,5353,5355],{},[2542,5354,3298],{},[2542,5356,4420],{},[2689,5358,573],{},[2542,5360,4420],{},[2542,5362,4959],{},[2542,5364,3298],{},[2549,5366,5367],{"encoding":2551},"t_n = n dt",[507,5369,5371,5428],{"className":5370,"ariaHidden":2557},[2556],[507,5372,5374,5378,5419,5422,5425],{"className":5373},[2561],[507,5375],{"className":5376,"style":5377},[2565],"height:0.7651em;vertical-align:-0.15em;",[507,5379,5381,5384],{"className":5380},[2570],[507,5382,3298],{"className":5383},[2570,2611],[507,5385,5387],{"className":5386},[2579],[507,5388,5390,5411],{"className":5389},[2583,3200],[507,5391,5393,5408],{"className":5392},[2587],[507,5394,5396],{"className":5395,"style":4507},[2591],[507,5397,5399,5402],{"style":5398},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[507,5400],{"className":5401,"style":2600},[2599],[507,5403,5405],{"className":5404},[2604,2605,2606,2607],[507,5406,4420],{"className":5407},[2570,2611,2607],[507,5409,3225],{"className":5410},[3224],[507,5412,5414],{"className":5413},[2587],[507,5415,5417],{"className":5416,"style":3232},[2591],[507,5418],{},[507,5420],{"className":5421,"style":2919},[2714],[507,5423,573],{"className":5424},[2923],[507,5426],{"className":5427,"style":2919},[2714],[507,5429,5431,5435,5438,5441],{"className":5430},[2561],[507,5432],{"className":5433,"style":5434},[2565],"height:0.6944em;",[507,5436,4420],{"className":5437},[2570,2611],[507,5439,4959],{"className":5440},[2570,2611],[507,5442,3298],{"className":5443},[2570,2611]," are the multiples of the time step ",[507,5446,5448,5464],{"className":5447},[2523],[507,5449,5451],{"className":5450},[2527],[2529,5452,5453],{"xmlns":2531},[2533,5454,5455,5461],{},[2536,5456,5457,5459],{},[2542,5458,4959],{},[2542,5460,3298],{},[2549,5462,5463],{"encoding":2551},"dt",[507,5465,5467],{"className":5466,"ariaHidden":2557},[2556],[507,5468,5470,5473,5476],{"className":5469},[2561],[507,5471],{"className":5472,"style":5434},[2565],[507,5474,4959],{"className":5475},[2570,2611],[507,5477,3298],{"className":5478},[2570,2611]," chosen. On a quantum computer, the calculation of each matrix elements can be done with any algorithm which allows to obtain overlap between quantum states. This tutorial focuses on the Hadamard test. Given that the ",[507,5481,5483,5500],{"className":5482},[2523],[507,5484,5486],{"className":5485},[2527],[2529,5487,5488],{"xmlns":2531},[2533,5489,5490,5498],{},[2536,5491,5492],{},[3168,5493,5494,5496],{},[2542,5495,2545],{"mathvariant":2544},[2542,5497,3279],{},[2549,5499,3393],{"encoding":2551},[507,5501,5503],{"className":5502,"ariaHidden":2557},[2556],[507,5504,5506,5509],{"className":5505},[2561],[507,5507],{"className":5508,"style":3187},[2565],[507,5510,5512,5515],{"className":5511},[2570],[507,5513,2545],{"className":5514,"style":2575},[2570,2574],[507,5516,5518],{"className":5517},[2579],[507,5519,5521,5541],{"className":5520},[2583,3200],[507,5522,5524,5538],{"className":5523},[2587],[507,5525,5527],{"className":5526,"style":3207},[2591],[507,5528,5529,5532],{"style":3210},[507,5530],{"className":5531,"style":2600},[2599],[507,5533,5535],{"className":5534},[2604,2605,2606,2607],[507,5536,3279],{"className":5537,"style":3314},[2570,2611,2607],[507,5539,3225],{"className":5540},[3224],[507,5542,5544],{"className":5543},[2587],[507,5545,5547],{"className":5546,"style":3232},[2591],[507,5548],{}," has dimension ",[507,5551,5553,5566],{"className":5552},[2523],[507,5554,5556],{"className":5555},[2527],[2529,5557,5558],{"xmlns":2531},[2533,5559,5560,5564],{},[2536,5561,5562],{},[2542,5563,2216],{},[2549,5565,2216],{"encoding":2551},[507,5567,5569],{"className":5568,"ariaHidden":2557},[2556],[507,5570,5572,5575],{"className":5571},[2561],[507,5573],{"className":5574,"style":2639},[2565],[507,5576,2216],{"className":5577,"style":2612},[2570,2611],", the Hamiltonian projected into the subspace will have dimensions ",[507,5580,5582,5601],{"className":5581},[2523],[507,5583,5585],{"className":5584},[2527],[2529,5586,5587],{"xmlns":2531},[2533,5588,5589,5598],{},[2536,5590,5591,5593,5596],{},[2542,5592,2216],{},[2689,5594,5595],{},"×",[2542,5597,2216],{},[2549,5599,5600],{"encoding":2551},"r \\times r",[507,5602,5604,5622],{"className":5603,"ariaHidden":2557},[2556],[507,5605,5607,5610,5613,5616,5619],{"className":5606},[2561],[507,5608],{"className":5609,"style":2707},[2565],[507,5611,2216],{"className":5612,"style":2612},[2570,2611],[507,5614],{"className":5615,"style":2715},[2714],[507,5617,5595],{"className":5618},[2719],[507,5620],{"className":5621,"style":2715},[2714],[507,5623,5625,5628],{"className":5624},[2561],[507,5626],{"className":5627,"style":2639},[2565],[507,5629,2216],{"className":5630,"style":2612},[2570,2611],". With ",[507,5633,5635,5648],{"className":5634},[2523],[507,5636,5638],{"className":5637},[2527],[2529,5639,5640],{"xmlns":2531},[2533,5641,5642,5646],{},[2536,5643,5644],{},[2542,5645,2216],{},[2549,5647,2216],{"encoding":2551},[507,5649,5651],{"className":5650,"ariaHidden":2557},[2556],[507,5652,5654,5657],{"className":5653},[2561],[507,5655],{"className":5656,"style":2639},[2565],[507,5658,2216],{"className":5659,"style":2612},[2570,2611]," small enough (generally ",[507,5662,5664,5686],{"className":5663},[2523],[507,5665,5667],{"className":5666},[2527],[2529,5668,5669],{"xmlns":2531},[2533,5670,5671,5683],{},[2536,5672,5673,5675,5678,5680],{},[2542,5674,2216],{},[2689,5676,5677],{},"\u003C",[2689,5679,5677],{},[2693,5681,5682],{},"100",[2549,5684,5685],{"encoding":2551},"r\u003C\u003C100",[507,5687,5689,5709],{"className":5688,"ariaHidden":2557},[2556],[507,5690,5692,5696,5699,5702,5706],{"className":5691},[2561],[507,5693],{"className":5694,"style":5695},[2565],"height:0.5782em;vertical-align:-0.0391em;",[507,5697,2216],{"className":5698,"style":2612},[2570,2611],[507,5700],{"className":5701,"style":2919},[2714],[507,5703,5705],{"className":5704},[2923],"\u003C\u003C",[507,5707],{"className":5708,"style":2919},[2714],[507,5710,5712,5715],{"className":5711},[2561],[507,5713],{"className":5714,"style":2729},[2565],[507,5716,5682],{"className":5717},[2570]," is sufficient to obtain convergence of estimates of eigenenergies) we can then easily diagonalize the projected Hamiltonian ",[507,5720,5722,5739],{"className":5721},[2523],[507,5723,5725],{"className":5724},[2527],[2529,5726,5727],{"xmlns":2531},[2533,5728,5729,5737],{},[2536,5730,5731],{},[4271,5732,5733,5735],{"accent":2557},[2542,5734,3138],{},[2689,5736,4277],{},[2549,5738,4280],{"encoding":2551},[507,5740,5742],{"className":5741,"ariaHidden":2557},[2556],[507,5743,5745,5748],{"className":5744},[2561],[507,5746],{"className":5747,"style":4290},[2565],[507,5749,5751],{"className":5750},[2570,4294],[507,5752,5754],{"className":5753},[2583],[507,5755,5757],{"className":5756},[2587],[507,5758,5760,5768],{"className":5759,"style":4290},[2591],[507,5761,5762,5765],{"style":4306},[507,5763],{"className":5764,"style":4310},[2599],[507,5766,3138],{"className":5767,"style":3153},[2570,2611],[507,5769,5770,5773],{"style":4316},[507,5771],{"className":5772,"style":4310},[2599],[507,5774,5776],{"className":5775,"style":4324},[4323],[507,5777,4277],{"className":5778},[2570],". However, we cannot directly diagonalize ",[507,5781,5783,5800],{"className":5782},[2523],[507,5784,5786],{"className":5785},[2527],[2529,5787,5788],{"xmlns":2531},[2533,5789,5790,5798],{},[2536,5791,5792],{},[4271,5793,5794,5796],{"accent":2557},[2542,5795,3138],{},[2689,5797,4277],{},[2549,5799,4280],{"encoding":2551},[507,5801,5803],{"className":5802,"ariaHidden":2557},[2556],[507,5804,5806,5809],{"className":5805},[2561],[507,5807],{"className":5808,"style":4290},[2565],[507,5810,5812],{"className":5811},[2570,4294],[507,5813,5815],{"className":5814},[2583],[507,5816,5818],{"className":5817},[2587],[507,5819,5821,5829],{"className":5820,"style":4290},[2591],[507,5822,5823,5826],{"style":4306},[507,5824],{"className":5825,"style":4310},[2599],[507,5827,3138],{"className":5828,"style":3153},[2570,2611],[507,5830,5831,5834],{"style":4316},[507,5832],{"className":5833,"style":4310},[2599],[507,5835,5837],{"className":5836,"style":4324},[4323],[507,5838,4277],{"className":5839},[2570]," because of the non-orthogonality of the Krylov space vectors. We'll have to measure their overlaps and construct a matrix ",[507,5842,5844,5863],{"className":5843},[2523],[507,5845,5847],{"className":5846},[2527],[2529,5848,5849],{"xmlns":2531},[2533,5850,5851,5860],{},[2536,5852,5853],{},[4271,5854,5855,5858],{"accent":2557},[2542,5856,5857],{},"S",[2689,5859,4277],{},[2549,5861,5862],{"encoding":2551},"\\tilde{S}",[507,5864,5866],{"className":5865,"ariaHidden":2557},[2556],[507,5867,5869,5872],{"className":5868},[2561],[507,5870],{"className":5871,"style":4290},[2565],[507,5873,5875],{"className":5874},[2570,4294],[507,5876,5878],{"className":5877},[2583],[507,5879,5881],{"className":5880},[2587],[507,5882,5884,5893],{"className":5883,"style":4290},[2591],[507,5885,5886,5889],{"style":4306},[507,5887],{"className":5888,"style":4310},[2599],[507,5890,5857],{"className":5891,"style":5892},[2570,2611],"margin-right:0.0576em;",[507,5894,5895,5898],{"style":4316},[507,5896],{"className":5897,"style":4310},[2599],[507,5899,5902],{"className":5900,"style":5901},[4323],"left:-0.1667em;",[507,5903,4277],{"className":5904},[2570],[507,5906,5908],{"className":5907},[2784],[507,5909,5911,5957],{"className":5910},[2523],[507,5912,5914],{"className":5913},[2527],[2529,5915,5916],{"xmlns":2531,"display":2793},[2533,5917,5918,5954],{},[2536,5919,5920,5934,5936,5938,5944,5946,5952],{},[3168,5921,5922,5928],{},[4271,5923,5924,5926],{"accent":2557},[2542,5925,5857],{},[2689,5927,4277],{},[2536,5929,5930,5932],{},[2542,5931,4417],{},[2542,5933,4420],{},[2689,5935,573],{},[2689,5937,4425],{"stretchy":2755},[3168,5939,5940,5942],{},[2542,5941,3793],{},[2542,5943,4417],{},[2542,5945,2749],{"mathvariant":2748},[3168,5947,5948,5950],{},[2542,5949,3793],{},[2542,5951,4420],{},[2689,5953,2756],{"stretchy":2755},[2549,5955,5956],{"encoding":2551},"\\tilde{S}_{mn} = \\langle \\psi_m \\vert \\psi_n \\rangle",[507,5958,5960,6047],{"className":5959,"ariaHidden":2557},[2556],[507,5961,5963,5966,6038,6041,6044],{"className":5962},[2561],[507,5964],{"className":5965,"style":4460},[2565],[507,5967,5969,6000],{"className":5968},[2570],[507,5970,5972],{"className":5971},[2570,4294],[507,5973,5975],{"className":5974},[2583],[507,5976,5978],{"className":5977},[2587],[507,5979,5981,5989],{"className":5980,"style":4290},[2591],[507,5982,5983,5986],{"style":4306},[507,5984],{"className":5985,"style":4310},[2599],[507,5987,5857],{"className":5988,"style":5892},[2570,2611],[507,5990,5991,5994],{"style":4316},[507,5992],{"className":5993,"style":4310},[2599],[507,5995,5997],{"className":5996,"style":5901},[4323],[507,5998,4277],{"className":5999},[2570],[507,6001,6003],{"className":6002},[2579],[507,6004,6006,6030],{"className":6005},[2583,3200],[507,6007,6009,6027],{"className":6008},[2587],[507,6010,6012],{"className":6011,"style":4507},[2591],[507,6013,6015,6018],{"style":6014},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[507,6016],{"className":6017,"style":2600},[2599],[507,6019,6021],{"className":6020},[2604,2605,2606,2607],[507,6022,6024],{"className":6023},[2570,2607],[507,6025,2693],{"className":6026},[2570,2611,2607],[507,6028,3225],{"className":6029},[3224],[507,6031,6033],{"className":6032},[2587],[507,6034,6036],{"className":6035,"style":3232},[2591],[507,6037],{},[507,6039],{"className":6040,"style":2919},[2714],[507,6042,573],{"className":6043},[2923],[507,6045],{"className":6046,"style":2919},[2714],[507,6048,6050,6053,6056,6096,6099,6139],{"className":6049},[2561],[507,6051],{"className":6052,"style":2769},[2565],[507,6054,4425],{"className":6055},[2941],[507,6057,6059,6062],{"className":6058},[2570],[507,6060,3793],{"className":6061,"style":2776},[2570,2611],[507,6063,6065],{"className":6064},[2579],[507,6066,6068,6088],{"className":6067},[2583,3200],[507,6069,6071,6085],{"className":6070},[2587],[507,6072,6074],{"className":6073,"style":4507},[2591],[507,6075,6076,6079],{"style":4572},[507,6077],{"className":6078,"style":2600},[2599],[507,6080,6082],{"className":6081},[2604,2605,2606,2607],[507,6083,4417],{"className":6084},[2570,2611,2607],[507,6086,3225],{"className":6087},[3224],[507,6089,6091],{"className":6090},[2587],[507,6092,6094],{"className":6093,"style":3232},[2591],[507,6095],{},[507,6097,2749],{"className":6098},[2570],[507,6100,6102,6105],{"className":6101},[2570],[507,6103,3793],{"className":6104,"style":2776},[2570,2611],[507,6106,6108],{"className":6107},[2579],[507,6109,6111,6131],{"className":6110},[2583,3200],[507,6112,6114,6128],{"className":6113},[2587],[507,6115,6117],{"className":6116,"style":4507},[2591],[507,6118,6119,6122],{"style":4572},[507,6120],{"className":6121,"style":2600},[2599],[507,6123,6125],{"className":6124},[2604,2605,2606,2607],[507,6126,4420],{"className":6127},[2570,2611,2607],[507,6129,3225],{"className":6130},[3224],[507,6132,6134],{"className":6133},[2587],[507,6135,6137],{"className":6136,"style":3232},[2591],[507,6138],{},[507,6140,2756],{"className":6141},[2780],[18,6143,6144],{},"This allows us to solve the eigenvalue problem in a non-orthogonal space (also called generalized eigenvalue problem)",[507,6146,6148],{"className":6147},[2784],[507,6149,6151,6202],{"className":6150},[2523],[507,6152,6154],{"className":6153},[2527],[2529,6155,6156],{"xmlns":2531,"display":2793},[2533,6157,6158,6199],{},[2536,6159,6160,6166,6170,6178,6180,6183,6185,6191,6193],{},[4271,6161,6162,6164],{"accent":2557},[2542,6163,3138],{},[2689,6165,4277],{},[6167,6168,6169],"mtext",{}," ",[4271,6171,6172,6175],{"accent":2557},[2542,6173,6174],{},"c",[2689,6176,6177],{},"⃗",[2689,6179,573],{},[2542,6181,6182],{},"E",[6167,6184,6169],{},[4271,6186,6187,6189],{"accent":2557},[2542,6188,5857],{},[2689,6190,4277],{},[6167,6192,6169],{},[4271,6194,6195,6197],{"accent":2557},[2542,6196,6174],{},[2689,6198,6177],{},[2549,6200,6201],{"encoding":2551},"\\tilde{H} \\ \\vec{c} = E \\ \\tilde{S} \\ \\vec{c}",[507,6203,6205,6302],{"className":6204,"ariaHidden":2557},[2556],[507,6206,6208,6211,6242,6245,6293,6296,6299],{"className":6207},[2561],[507,6209],{"className":6210,"style":4290},[2565],[507,6212,6214],{"className":6213},[2570,4294],[507,6215,6217],{"className":6216},[2583],[507,6218,6220],{"className":6219},[2587],[507,6221,6223,6231],{"className":6222,"style":4290},[2591],[507,6224,6225,6228],{"style":4306},[507,6226],{"className":6227,"style":4310},[2599],[507,6229,3138],{"className":6230,"style":3153},[2570,2611],[507,6232,6233,6236],{"style":4316},[507,6234],{"className":6235,"style":4310},[2599],[507,6237,6239],{"className":6238,"style":4324},[4323],[507,6240,4277],{"className":6241},[2570],[507,6243,6169],{"className":6244},[2714],[507,6246,6248],{"className":6247},[2570,4294],[507,6249,6251],{"className":6250},[2583],[507,6252,6254],{"className":6253},[2587],[507,6255,6258,6266],{"className":6256,"style":6257},[2591],"height:0.714em;",[507,6259,6260,6263],{"style":4306},[507,6261],{"className":6262,"style":4310},[2599],[507,6264,6174],{"className":6265},[2570,2611],[507,6267,6268,6271],{"style":4306},[507,6269],{"className":6270,"style":4310},[2599],[507,6272,6275],{"className":6273,"style":6274},[4323],"left:-0.1799em;",[507,6276,6280],{"className":6277,"style":6279},[6278],"overlay","height:0.714em;width:0.471em;",[6281,6282,6289],"svg",{"xmlns":6283,"width":6284,"height":6285,"style":6286,"viewBox":6287,"preserveAspectRatio":6288},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.471em","0.714em","width:0.471em","0 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can then obtain estimates of the eigenvalues and eigenstates of ",[507,6387,6389,6402],{"className":6388},[2523],[507,6390,6392],{"className":6391},[2527],[2529,6393,6394],{"xmlns":2531},[2533,6395,6396,6400],{},[2536,6397,6398],{},[2542,6399,3138],{},[2549,6401,3138],{"encoding":2551},[507,6403,6405],{"className":6404,"ariaHidden":2557},[2556],[507,6406,6408,6411],{"className":6407},[2561],[507,6409],{"className":6410,"style":2566},[2565],[507,6412,3138],{"className":6413,"style":3153},[2570,2611]," by looking at the ones of ",[507,6416,6418,6435],{"className":6417},[2523],[507,6419,6421],{"className":6420},[2527],[2529,6422,6423],{"xmlns":2531},[2533,6424,6425,6433],{},[2536,6426,6427],{},[4271,6428,6429,6431],{"accent":2557},[2542,6430,3138],{},[2689,6432,4277],{},[2549,6434,4280],{"encoding":2551},[507,6436,6438],{"className":6437,"ariaHidden":2557},[2556],[507,6439,6441,6444],{"className":6440},[2561],[507,6442],{"className":6443,"style":4290},[2565],[507,6445,6447],{"className":6446},[2570,4294],[507,6448,6450],{"className":6449},[2583],[507,6451,6453],{"className":6452},[2587],[507,6454,6456,6464],{"className":6455,"style":4290},[2591],[507,6457,6458,6461],{"style":4306},[507,6459],{"className":6460,"style":4310},[2599],[507,6462,3138],{"className":6463,"style":3153},[2570,2611],[507,6465,6466,6469],{"style":4316},[507,6467],{"className":6468,"style":4310},[2599],[507,6470,6472],{"className":6471,"style":4324},[4323],[507,6473,4277],{"className":6474},[2570],". For example, the estimate of the ground state energy is obtained by taking the smallest eigenvalue ",[507,6477,6479,6492],{"className":6478},[2523],[507,6480,6482],{"className":6481},[2527],[2529,6483,6484],{"xmlns":2531},[2533,6485,6486,6490],{},[2536,6487,6488],{},[2542,6489,6174],{},[2549,6491,6174],{"encoding":2551},[507,6493,6495],{"className":6494,"ariaHidden":2557},[2556],[507,6496,6498,6501],{"className":6497},[2561],[507,6499],{"className":6500,"style":2639},[2565],[507,6502,6174],{"className":6503},[2570,2611]," and the ground state from the corresponding eigenvector ",[507,6506,6508,6526],{"className":6507},[2523],[507,6509,6511],{"className":6510},[2527],[2529,6512,6513],{"xmlns":2531},[2533,6514,6515,6523],{},[2536,6516,6517],{},[4271,6518,6519,6521],{"accent":2557},[2542,6520,6174],{},[2689,6522,6177],{},[2549,6524,6525],{"encoding":2551},"\\vec{c}",[507,6527,6529],{"className":6528,"ariaHidden":2557},[2556],[507,6530,6532,6535],{"className":6531},[2561],[507,6533],{"className":6534,"style":6257},[2565],[507,6536,6538],{"className":6537},[2570,4294],[507,6539,6541],{"className":6540},[2583],[507,6542,6544],{"className":6543},[2587],[507,6545,6547,6555],{"className":6546,"style":6257},[2591],[507,6548,6549,6552],{"style":4306},[507,6550],{"className":6551,"style":4310},[2599],[507,6553,6174],{"className":6554},[2570,2611],[507,6556,6557,6560],{"style":4306},[507,6558],{"className":6559,"style":4310},[2599],[507,6561,6563],{"className":6562,"style":6274},[4323],[507,6564,6566],{"className":6565,"style":6279},[6278],[6281,6567,6568],{"xmlns":6283,"width":6284,"height":6285,"style":6286,"viewBox":6287,"preserveAspectRatio":6288},[6290,6569],{"d":6292},". The coefficients in ",[507,6572,6574,6591],{"className":6573},[2523],[507,6575,6577],{"className":6576},[2527],[2529,6578,6579],{"xmlns":2531},[2533,6580,6581,6589],{},[2536,6582,6583],{},[4271,6584,6585,6587],{"accent":2557},[2542,6586,6174],{},[2689,6588,6177],{},[2549,6590,6525],{"encoding":2551},[507,6592,6594],{"className":6593,"ariaHidden":2557},[2556],[507,6595,6597,6600],{"className":6596},[2561],[507,6598],{"className":6599,"style":6257},[2565],[507,6601,6603],{"className":6602},[2570,4294],[507,6604,6606],{"className":6605},[2583],[507,6607,6609],{"className":6608},[2587],[507,6610,6612,6620],{"className":6611,"style":6257},[2591],[507,6613,6614,6617],{"style":4306},[507,6615],{"className":6616,"style":4310},[2599],[507,6618,6174],{"className":6619},[2570,2611],[507,6621,6622,6625],{"style":4306},[507,6623],{"className":6624,"style":4310},[2599],[507,6626,6628],{"className":6627,"style":6274},[4323],[507,6629,6631],{"className":6630,"style":6279},[6278],[6281,6632,6633],{"xmlns":6283,"width":6284,"height":6285,"style":6286,"viewBox":6287,"preserveAspectRatio":6288},[6290,6634],{"d":6292}," determines the contribution of the different vectors that span ",[507,6637,6639,6656],{"className":6638},[2523],[507,6640,6642],{"className":6641},[2527],[2529,6643,6644],{"xmlns":2531},[2533,6645,6646,6654],{},[2536,6647,6648],{},[3168,6649,6650,6652],{},[2542,6651,2545],{"mathvariant":2544},[2542,6653,3279],{},[2549,6655,3393],{"encoding":2551},[507,6657,6659],{"className":6658,"ariaHidden":2557},[2556],[507,6660,6662,6665],{"className":6661},[2561],[507,6663],{"className":6664,"style":3187},[2565],[507,6666,6668,6671],{"className":6667},[2570],[507,6669,2545],{"className":6670,"style":2575},[2570,2574],[507,6672,6674],{"className":6673},[2579],[507,6675,6677,6697],{"className":6676},[2583,3200],[507,6678,6680,6694],{"className":6679},[2587],[507,6681,6683],{"className":6682,"style":3207},[2591],[507,6684,6685,6688],{"style":3210},[507,6686],{"className":6687,"style":2600},[2599],[507,6689,6691],{"className":6690},[2604,2605,2606,2607],[507,6692,3279],{"className":6693,"style":3314},[2570,2611,2607],[507,6695,3225],{"className":6696},[3224],[507,6698,6700],{"className":6699},[2587],[507,6701,6703],{"className":6702,"style":3232},[2591],[507,6704],{},[18,6706,6707],{},[6708,6709],"img",{"alt":6710,"src":6711},"fig1.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-06.avif",[18,6713,6714,6715,6836,6837,622,6917,6998,6999,622,7086,7170,7171,7231,7232,7314,7315,7394,7395,7477,7478,7506,7507,7658],{},"The Figure shows a circuit representation of the modified Hadamard test, a method that is used to compute the overlap between different quantum states. For each matrix element ",[507,6716,6718,6746],{"className":6717},[2523],[507,6719,6721],{"className":6720},[2527],[2529,6722,6723],{"xmlns":2531},[2533,6724,6725,6743],{},[2536,6726,6727],{},[3168,6728,6729,6735],{},[4271,6730,6731,6733],{"accent":2557},[2542,6732,3138],{},[2689,6734,4277],{},[2536,6736,6737,6739,6741],{},[2542,6738,3293],{},[2689,6740,2819],{"separator":2557},[2542,6742,2372],{},[2549,6744,6745],{"encoding":2551},"\\tilde{H}_{i,j}",[507,6747,6749],{"className":6748,"ariaHidden":2557},[2556],[507,6750,6752,6756],{"className":6751},[2561],[507,6753],{"className":6754,"style":6755},[2565],"height:1.2063em;vertical-align:-0.2861em;",[507,6757,6759,6790],{"className":6758},[2570],[507,6760,6762],{"className":6761},[2570,4294],[507,6763,6765],{"className":6764},[2583],[507,6766,6768],{"className":6767},[2587],[507,6769,6771,6779],{"className":6770,"style":4290},[2591],[507,6772,6773,6776],{"style":4306},[507,6774],{"className":6775,"style":4310},[2599],[507,6777,3138],{"className":6778,"style":3153},[2570,2611],[507,6780,6781,6784],{"style":4316},[507,6782],{"className":6783,"style":4310},[2599],[507,6785,6787],{"className":6786,"style":4324},[4323],[507,6788,4277],{"className":6789},[2570],[507,6791,6793],{"className":6792},[2579],[507,6794,6796,6827],{"className":6795},[2583,3200],[507,6797,6799,6824],{"className":6798},[2587],[507,6800,6803],{"className":6801,"style":6802},[2591],"height:0.3117em;",[507,6804,6805,6808],{"style":4510},[507,6806],{"className":6807,"style":2600},[2599],[507,6809,6811],{"className":6810},[2604,2605,2606,2607],[507,6812,6814,6817,6820],{"className":6813},[2570,2607],[507,6815,3293],{"className":6816},[2570,2611,2607],[507,6818,2819],{"className":6819},[2961,2607],[507,6821,2372],{"className":6822,"style":6823},[2570,2611,2607],"margin-right:0.0572em;",[507,6825,3225],{"className":6826},[3224],[507,6828,6830],{"className":6829},[2587],[507,6831,6834],{"className":6832,"style":6833},[2591],"height:0.2861em;",[507,6835],{},", a Hadamard test between the state ",[507,6838,6840,6862],{"className":6839},[2523],[507,6841,6843],{"className":6842},[2527],[2529,6844,6845],{"xmlns":2531},[2533,6846,6847,6859],{},[2536,6848,6849,6851,6857],{},[2542,6850,2749],{"mathvariant":2748},[3168,6852,6853,6855],{},[2542,6854,3793],{},[2542,6856,3293],{},[2689,6858,2756],{"stretchy":2755},[2549,6860,6861],{"encoding":2551},"\\vert \\psi_i \\rangle",[507,6863,6865],{"className":6864,"ariaHidden":2557},[2556],[507,6866,6868,6871,6874,6914],{"className":6867},[2561],[507,6869],{"className":6870,"style":2769},[2565],[507,6872,2749],{"className":6873},[2570],[507,6875,6877,6880],{"className":6876},[2570],[507,6878,3793],{"className":6879,"style":2776},[2570,2611],[507,6881,6883],{"className":6882},[2579],[507,6884,6886,6906],{"className":6885},[2583,3200],[507,6887,6889,6903],{"className":6888},[2587],[507,6890,6892],{"className":6891,"style":6802},[2591],[507,6893,6894,6897],{"style":4572},[507,6895],{"className":6896,"style":2600},[2599],[507,6898,6900],{"className":6899},[2604,2605,2606,2607],[507,6901,3293],{"className":6902},[2570,2611,2607],[507,6904,3225],{"className":6905},[3224],[507,6907,6909],{"className":6908},[2587],[507,6910,6912],{"className":6911,"style":3232},[2591],[507,6913],{},[507,6915,2756],{"className":6916},[2780],[507,6918,6920,6942],{"className":6919},[2523],[507,6921,6923],{"className":6922},[2527],[2529,6924,6925],{"xmlns":2531},[2533,6926,6927,6939],{},[2536,6928,6929,6931,6937],{},[2542,6930,2749],{"mathvariant":2748},[3168,6932,6933,6935],{},[2542,6934,3793],{},[2542,6936,2372],{},[2689,6938,2756],{"stretchy":2755},[2549,6940,6941],{"encoding":2551},"\\vert \\psi_j \\rangle",[507,6943,6945],{"className":6944,"ariaHidden":2557},[2556],[507,6946,6948,6952,6955,6995],{"className":6947},[2561],[507,6949],{"className":6950,"style":6951},[2565],"height:1.0361em;vertical-align:-0.2861em;",[507,6953,2749],{"className":6954},[2570],[507,6956,6958,6961],{"className":6957},[2570],[507,6959,3793],{"className":6960,"style":2776},[2570,2611],[507,6962,6964],{"className":6963},[2579],[507,6965,6967,6987],{"className":6966},[2583,3200],[507,6968,6970,6984],{"className":6969},[2587],[507,6971,6973],{"className":6972,"style":6802},[2591],[507,6974,6975,6978],{"style":4572},[507,6976],{"className":6977,"style":2600},[2599],[507,6979,6981],{"className":6980},[2604,2605,2606,2607],[507,6982,2372],{"className":6983,"style":6823},[2570,2611,2607],[507,6985,3225],{"className":6986},[3224],[507,6988,6990],{"className":6989},[2587],[507,6991,6993],{"className":6992,"style":6833},[2591],[507,6994],{},[507,6996,2756],{"className":6997},[2780]," is carried out. This is highlighted in the figure by the color scheme for the matrix elements and the corresponding ",[507,7000,7002,7026],{"className":7001},[2523],[507,7003,7005],{"className":7004},[2527],[2529,7006,7007],{"xmlns":2531},[2533,7008,7009,7023],{},[2536,7010,7011,7014,7017],{},[6167,7012,7013],{},"Prep",[6167,7015,7016],{},"  ",[3168,7018,7019,7021],{},[2542,7020,3793],{},[2542,7022,3293],{},[2549,7024,7025],{"encoding":2551},"\\text{Prep} \\; \\psi_i",[507,7027,7029],{"className":7028,"ariaHidden":2557},[2556],[507,7030,7032,7036,7043,7046],{"className":7031},[2561],[507,7033],{"className":7034,"style":7035},[2565],"height:0.8889em;vertical-align:-0.1944em;",[507,7037,7040],{"className":7038},[2570,7039],"text",[507,7041,7013],{"className":7042},[2570],[507,7044],{"className":7045,"style":2919},[2714],[507,7047,7049,7052],{"className":7048},[2570],[507,7050,3793],{"className":7051,"style":2776},[2570,2611],[507,7053,7055],{"className":7054},[2579],[507,7056,7058,7078],{"className":7057},[2583,3200],[507,7059,7061,7075],{"className":7060},[2587],[507,7062,7064],{"className":7063,"style":6802},[2591],[507,7065,7066,7069],{"style":4572},[507,7067],{"className":7068,"style":2600},[2599],[507,7070,7072],{"className":7071},[2604,2605,2606,2607],[507,7073,3293],{"className":7074},[2570,2611,2607],[507,7076,3225],{"className":7077},[3224],[507,7079,7081],{"className":7080},[2587],[507,7082,7084],{"className":7083,"style":3232},[2591],[507,7085],{},[507,7087,7089,7111],{"className":7088},[2523],[507,7090,7092],{"className":7091},[2527],[2529,7093,7094],{"xmlns":2531},[2533,7095,7096,7108],{},[2536,7097,7098,7100,7102],{},[6167,7099,7013],{},[6167,7101,7016],{},[3168,7103,7104,7106],{},[2542,7105,3793],{},[2542,7107,2372],{},[2549,7109,7110],{"encoding":2551},"\\text{Prep} \\; \\psi_j",[507,7112,7114],{"className":7113,"ariaHidden":2557},[2556],[507,7115,7117,7121,7127,7130],{"className":7116},[2561],[507,7118],{"className":7119,"style":7120},[2565],"height:0.9805em;vertical-align:-0.2861em;",[507,7122,7124],{"className":7123},[2570,7039],[507,7125,7013],{"className":7126},[2570],[507,7128],{"className":7129,"style":2919},[2714],[507,7131,7133,7136],{"className":7132},[2570],[507,7134,3793],{"className":7135,"style":2776},[2570,2611],[507,7137,7139],{"className":7138},[2579],[507,7140,7142,7162],{"className":7141},[2583,3200],[507,7143,7145,7159],{"className":7144},[2587],[507,7146,7148],{"className":7147,"style":6802},[2591],[507,7149,7150,7153],{"style":4572},[507,7151],{"className":7152,"style":2600},[2599],[507,7154,7156],{"className":7155},[2604,2605,2606,2607],[507,7157,2372],{"className":7158,"style":6823},[2570,2611,2607],[507,7160,3225],{"className":7161},[3224],[507,7163,7165],{"className":7164},[2587],[507,7166,7168],{"className":7167,"style":6833},[2591],[507,7169],{}," operations. Thus, a set of Hadamard tests for all the possible combinations of Krylov space vectors is required to compute all the matrix elements of the projected Hamiltonian ",[507,7172,7174,7191],{"className":7173},[2523],[507,7175,7177],{"className":7176},[2527],[2529,7178,7179],{"xmlns":2531},[2533,7180,7181,7189],{},[2536,7182,7183],{},[4271,7184,7185,7187],{"accent":2557},[2542,7186,3138],{},[2689,7188,4277],{},[2549,7190,4280],{"encoding":2551},[507,7192,7194],{"className":7193,"ariaHidden":2557},[2556],[507,7195,7197,7200],{"className":7196},[2561],[507,7198],{"className":7199,"style":4290},[2565],[507,7201,7203],{"className":7202},[2570,4294],[507,7204,7206],{"className":7205},[2583],[507,7207,7209],{"className":7208},[2587],[507,7210,7212,7220],{"className":7211,"style":4290},[2591],[507,7213,7214,7217],{"style":4306},[507,7215],{"className":7216,"style":4310},[2599],[507,7218,3138],{"className":7219,"style":3153},[2570,2611],[507,7221,7222,7225],{"style":4316},[507,7223],{"className":7224,"style":4310},[2599],[507,7226,7228],{"className":7227,"style":4324},[4323],[507,7229,4277],{"className":7230},[2570],". The top wire in the Hadamard test circuit is an ancilla qubit which is measured either in the X or Y basis, its expectation value determines the value of the overlap between the states. The bottom wire represents all the qubits of the system Hamiltonian. The ",[507,7233,7235,7256],{"className":7234},[2523],[507,7236,7238],{"className":7237},[2527],[2529,7239,7240],{"xmlns":2531},[2533,7241,7242,7254],{},[2536,7243,7244,7246,7248],{},[6167,7245,7013],{},[6167,7247,7016],{},[3168,7249,7250,7252],{},[2542,7251,3793],{},[2542,7253,3293],{},[2549,7255,7025],{"encoding":2551},[507,7257,7259],{"className":7258,"ariaHidden":2557},[2556],[507,7260,7262,7265,7271,7274],{"className":7261},[2561],[507,7263],{"className":7264,"style":7035},[2565],[507,7266,7268],{"className":7267},[2570,7039],[507,7269,7013],{"className":7270},[2570],[507,7272],{"className":7273,"style":2919},[2714],[507,7275,7277,7280],{"className":7276},[2570],[507,7278,3793],{"className":7279,"style":2776},[2570,2611],[507,7281,7283],{"className":7282},[2579],[507,7284,7286,7306],{"className":7285},[2583,3200],[507,7287,7289,7303],{"className":7288},[2587],[507,7290,7292],{"className":7291,"style":6802},[2591],[507,7293,7294,7297],{"style":4572},[507,7295],{"className":7296,"style":2600},[2599],[507,7298,7300],{"className":7299},[2604,2605,2606,2607],[507,7301,3293],{"className":7302},[2570,2611,2607],[507,7304,3225],{"className":7305},[3224],[507,7307,7309],{"className":7308},[2587],[507,7310,7312],{"className":7311,"style":3232},[2591],[507,7313],{}," operation prepares the system qubit in the state ",[507,7316,7318,7339],{"className":7317},[2523],[507,7319,7321],{"className":7320},[2527],[2529,7322,7323],{"xmlns":2531},[2533,7324,7325,7337],{},[2536,7326,7327,7329,7335],{},[2542,7328,2749],{"mathvariant":2748},[3168,7330,7331,7333],{},[2542,7332,3793],{},[2542,7334,3293],{},[2689,7336,2756],{"stretchy":2755},[2549,7338,6861],{"encoding":2551},[507,7340,7342],{"className":7341,"ariaHidden":2557},[2556],[507,7343,7345,7348,7351,7391],{"className":7344},[2561],[507,7346],{"className":7347,"style":2769},[2565],[507,7349,2749],{"className":7350},[2570],[507,7352,7354,7357],{"className":7353},[2570],[507,7355,3793],{"className":7356,"style":2776},[2570,2611],[507,7358,7360],{"className":7359},[2579],[507,7361,7363,7383],{"className":7362},[2583,3200],[507,7364,7366,7380],{"className":7365},[2587],[507,7367,7369],{"className":7368,"style":6802},[2591],[507,7370,7371,7374],{"style":4572},[507,7372],{"className":7373,"style":2600},[2599],[507,7375,7377],{"className":7376},[2604,2605,2606,2607],[507,7378,3293],{"className":7379},[2570,2611,2607],[507,7381,3225],{"className":7382},[3224],[507,7384,7386],{"className":7385},[2587],[507,7387,7389],{"className":7388,"style":3232},[2591],[507,7390],{},[507,7392,2756],{"className":7393},[2780]," controlled by the state of the ancilla qubit (similarly for ",[507,7396,7398,7419],{"className":7397},[2523],[507,7399,7401],{"className":7400},[2527],[2529,7402,7403],{"xmlns":2531},[2533,7404,7405,7417],{},[2536,7406,7407,7409,7411],{},[6167,7408,7013],{},[6167,7410,7016],{},[3168,7412,7413,7415],{},[2542,7414,3793],{},[2542,7416,2372],{},[2549,7418,7110],{"encoding":2551},[507,7420,7422],{"className":7421,"ariaHidden":2557},[2556],[507,7423,7425,7428,7434,7437],{"className":7424},[2561],[507,7426],{"className":7427,"style":7120},[2565],[507,7429,7431],{"className":7430},[2570,7039],[507,7432,7013],{"className":7433},[2570],[507,7435],{"className":7436,"style":2919},[2714],[507,7438,7440,7443],{"className":7439},[2570],[507,7441,3793],{"className":7442,"style":2776},[2570,2611],[507,7444,7446],{"className":7445},[2579],[507,7447,7449,7469],{"className":7448},[2583,3200],[507,7450,7452,7466],{"className":7451},[2587],[507,7453,7455],{"className":7454,"style":6802},[2591],[507,7456,7457,7460],{"style":4572},[507,7458],{"className":7459,"style":2600},[2599],[507,7461,7463],{"className":7462},[2604,2605,2606,2607],[507,7464,2372],{"className":7465,"style":6823},[2570,2611,2607],[507,7467,3225],{"className":7468},[3224],[507,7470,7472],{"className":7471},[2587],[507,7473,7475],{"className":7474,"style":6833},[2591],[507,7476],{},") and the operation ",[507,7479,7481,7494],{"className":7480},[2523],[507,7482,7484],{"className":7483},[2527],[2529,7485,7486],{"xmlns":2531},[2533,7487,7488,7492],{},[2536,7489,7490],{},[2542,7491,3174],{},[2549,7493,3174],{"encoding":2551},[507,7495,7497],{"className":7496,"ariaHidden":2557},[2556],[507,7498,7500,7503],{"className":7499},[2561],[507,7501],{"className":7502,"style":2566},[2565],[507,7504,3174],{"className":7505,"style":3220},[2570,2611]," represents Pauli decomposition of the system Hamiltonian ",[507,7508,7510,7539],{"className":7509},[2523],[507,7511,7513],{"className":7512},[2527],[2529,7514,7515],{"xmlns":2531},[2533,7516,7517,7536],{},[2536,7518,7519,7521,7523,7530],{},[2542,7520,3138],{},[2689,7522,573],{},[3168,7524,7525,7528],{},[2689,7526,7527],{},"∑",[2542,7529,3293],{},[3168,7531,7532,7534],{},[2542,7533,3174],{},[2542,7535,3293],{},[2549,7537,7538],{"encoding":2551},"H = \\sum_i P_i",[507,7540,7542,7560],{"className":7541,"ariaHidden":2557},[2556],[507,7543,7545,7548,7551,7554,7557],{"className":7544},[2561],[507,7546],{"className":7547,"style":2566},[2565],[507,7549,3138],{"className":7550,"style":3153},[2570,2611],[507,7552],{"className":7553,"style":2919},[2714],[507,7555,573],{"className":7556},[2923],[507,7558],{"className":7559,"style":2919},[2714],[507,7561,7563,7567,7614,7617],{"className":7562},[2561],[507,7564],{"className":7565,"style":7566},[2565],"height:1.0497em;vertical-align:-0.2997em;",[507,7568,7571,7577],{"className":7569},[7570],"mop",[507,7572,7527],{"className":7573,"style":7576},[7570,7574,7575],"op-symbol","small-op","position:relative;top:0em;",[507,7578,7580],{"className":7579},[2579],[507,7581,7583,7605],{"className":7582},[2583,3200],[507,7584,7586,7602],{"className":7585},[2587],[507,7587,7590],{"className":7588,"style":7589},[2591],"height:0.162em;",[507,7591,7593,7596],{"style":7592},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[507,7594],{"className":7595,"style":2600},[2599],[507,7597,7599],{"className":7598},[2604,2605,2606,2607],[507,7600,3293],{"className":7601},[2570,2611,2607],[507,7603,3225],{"className":7604},[3224],[507,7606,7608],{"className":7607},[2587],[507,7609,7612],{"className":7610,"style":7611},[2591],"height:0.2997em;",[507,7613],{},[507,7615],{"className":7616,"style":2965},[2714],[507,7618,7620,7623],{"className":7619},[2570],[507,7621,3174],{"className":7622,"style":3220},[2570,2611],[507,7624,7626],{"className":7625},[2579],[507,7627,7629,7650],{"className":7628},[2583,3200],[507,7630,7632,7647],{"className":7631},[2587],[507,7633,7635],{"className":7634,"style":6802},[2591],[507,7636,7638,7641],{"style":7637},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[507,7639],{"className":7640,"style":2600},[2599],[507,7642,7644],{"className":7643},[2604,2605,2606,2607],[507,7645,3293],{"className":7646},[2570,2611,2607],[507,7648,3225],{"className":7649},[3224],[507,7651,7653],{"className":7652},[2587],[507,7654,7656],{"className":7655,"style":3232},[2591],[507,7657],{},". A more detailed derivation of the operations calculated by the Hadamard test is given below.",[7660,7661,7663],"h4",{"id":7662},"define-hamiltonian","Define Hamiltonian",[18,7665,7666,7667,7696,7697],{},"Let's consider the Heisenberg Hamiltonian for ",[507,7668,7670,7684],{"className":7669},[2523],[507,7671,7673],{"className":7672},[2527],[2529,7674,7675],{"xmlns":2531},[2533,7676,7677,7682],{},[2536,7678,7679],{},[2542,7680,7681],{},"N",[2549,7683,7681],{"encoding":2551},[507,7685,7687],{"className":7686,"ariaHidden":2557},[2556],[507,7688,7690,7693],{"className":7689},[2561],[507,7691],{"className":7692,"style":2566},[2565],[507,7694,7681],{"className":7695,"style":3314},[2570,2611]," qubits on a linear chain: ",[507,7698,7700,7776],{"className":7699},[2523],[507,7701,7703],{"className":7702},[2527],[2529,7704,7705],{"xmlns":2531},[2533,7706,7707,7773],{},[2536,7708,7709,7711,7713,7727,7734,7740,7742,7749,7755,7757,7760,7767],{},[2542,7710,3138],{},[2689,7712,573],{},[3775,7714,7715,7717,7725],{},[2689,7716,7527],{},[2536,7718,7719,7721,7723],{},[2542,7720,3293],{},[2689,7722,2819],{"separator":2557},[2542,7724,2372],{},[2542,7726,7681],{},[3168,7728,7729,7732],{},[2542,7730,7731],{},"X",[2542,7733,3293],{},[3168,7735,7736,7738],{},[2542,7737,7731],{},[2542,7739,2372],{},[2689,7741,2107],{},[3168,7743,7744,7747],{},[2542,7745,7746],{},"Y",[2542,7748,3293],{},[3168,7750,7751,7753],{},[2542,7752,7746],{},[2542,7754,2372],{},[2689,7756,2691],{},[2542,7758,7759],{},"J",[3168,7761,7762,7765],{},[2542,7763,7764],{},"Z",[2542,7766,3293],{},[3168,7768,7769,7771],{},[2542,7770,7764],{},[2542,7772,2372],{},[2549,7774,7775],{"encoding":2551},"H= \\sum_{i,j}^N X_i X_j + Y_i Y_j - J Z_i Z_j",[507,7777,7779,7797,7961,8058],{"className":7778,"ariaHidden":2557},[2556],[507,7780,7782,7785,7788,7791,7794],{"className":7781},[2561],[507,7783],{"className":7784,"style":2566},[2565],[507,7786,3138],{"className":7787,"style":3153},[2570,2611],[507,7789],{"className":7790,"style":2919},[2714],[507,7792,573],{"className":7793},[2923],[507,7795],{"className":7796,"style":2919},[2714],[507,7798,7800,7804,7867,7870,7912,7952,7955,7958],{"className":7799},[2561],[507,7801],{"className":7802,"style":7803},[2565],"height:1.417em;vertical-align:-0.4358em;",[507,7805,7807,7810],{"className":7806},[7570],[507,7808,7527],{"className":7809,"style":7576},[7570,7574,7575],[507,7811,7813],{"className":7812},[2579],[507,7814,7816,7858],{"className":7815},[2583,3200],[507,7817,7819,7855],{"className":7818},[2587],[507,7820,7823,7843],{"className":7821,"style":7822},[2591],"height:0.9812em;",[507,7824,7825,7828],{"style":7592},[507,7826],{"className":7827,"style":2600},[2599],[507,7829,7831],{"className":7830},[2604,2605,2606,2607],[507,7832,7834,7837,7840],{"className":7833},[2570,2607],[507,7835,3293],{"className":7836},[2570,2611,2607],[507,7838,2819],{"className":7839},[2961,2607],[507,7841,2372],{"className":7842,"style":6823},[2570,2611,2607],[507,7844,7846,7849],{"style":7845},"top:-3.2029em;margin-right:0.05em;",[507,7847],{"className":7848,"style":2600},[2599],[507,7850,7852],{"className":7851},[2604,2605,2606,2607],[507,7853,7681],{"className":7854,"style":3314},[2570,2611,2607],[507,7856,3225],{"className":7857},[3224],[507,7859,7861],{"className":7860},[2587],[507,7862,7865],{"className":7863,"style":7864},[2591],"height:0.4358em;",[507,7866],{},[507,7868],{"className":7869,"style":2965},[2714],[507,7871,7873,7877],{"className":7872},[2570],[507,7874,7731],{"className":7875,"style":7876},[2570,2611],"margin-right:0.0785em;",[507,7878,7880],{"className":7879},[2579],[507,7881,7883,7904],{"className":7882},[2583,3200],[507,7884,7886,7901],{"className":7885},[2587],[507,7887,7889],{"className":7888,"style":6802},[2591],[507,7890,7892,7895],{"style":7891},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[507,7893],{"className":7894,"style":2600},[2599],[507,7896,7898],{"className":7897},[2604,2605,2606,2607],[507,7899,3293],{"className":7900},[2570,2611,2607],[507,7902,3225],{"className":7903},[3224],[507,7905,7907],{"className":7906},[2587],[507,7908,7910],{"className":7909,"style":3232},[2591],[507,7911],{},[507,7913,7915,7918],{"className":7914},[2570],[507,7916,7731],{"className":7917,"style":7876},[2570,2611],[507,7919,7921],{"className":7920},[2579],[507,7922,7924,7944],{"className":7923},[2583,3200],[507,7925,7927,7941],{"className":7926},[2587],[507,7928,7930],{"className":7929,"style":6802},[2591],[507,7931,7932,7935],{"style":7891},[507,7933],{"className":7934,"style":2600},[2599],[507,7936,7938],{"className":7937},[2604,2605,2606,2607],[507,7939,2372],{"className":7940,"style":6823},[2570,2611,2607],[507,7942,3225],{"className":7943},[3224],[507,7945,7947],{"className":7946},[2587],[507,7948,7950],{"className":7949,"style":6833},[2591],[507,7951],{},[507,7953],{"className":7954,"style":2715},[2714],[507,7956,2107],{"className":7957},[2719],[507,7959],{"className":7960,"style":2715},[2714],[507,7962,7964,7968,8009,8049,8052,8055],{"className":7963},[2561],[507,7965],{"className":7966,"style":7967},[2565],"height:0.9694em;vertical-align:-0.2861em;",[507,7969,7971,7974],{"className":7970},[2570],[507,7972,7746],{"className":7973,"style":2715},[2570,2611],[507,7975,7977],{"className":7976},[2579],[507,7978,7980,8001],{"className":7979},[2583,3200],[507,7981,7983,7998],{"className":7982},[2587],[507,7984,7986],{"className":7985,"style":6802},[2591],[507,7987,7989,7992],{"style":7988},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[507,7990],{"className":7991,"style":2600},[2599],[507,7993,7995],{"className":7994},[2604,2605,2606,2607],[507,7996,3293],{"className":7997},[2570,2611,2607],[507,7999,3225],{"className":8000},[3224],[507,8002,8004],{"className":8003},[2587],[507,8005,8007],{"className":8006,"style":3232},[2591],[507,8008],{},[507,8010,8012,8015],{"className":8011},[2570],[507,8013,7746],{"className":8014,"style":2715},[2570,2611],[507,8016,8018],{"className":8017},[2579],[507,8019,8021,8041],{"className":8020},[2583,3200],[507,8022,8024,8038],{"className":8023},[2587],[507,8025,8027],{"className":8026,"style":6802},[2591],[507,8028,8029,8032],{"style":7988},[507,8030],{"className":8031,"style":2600},[2599],[507,8033,8035],{"className":8034},[2604,2605,2606,2607],[507,8036,2372],{"className":8037,"style":6823},[2570,2611,2607],[507,8039,3225],{"className":8040},[3224],[507,8042,8044],{"className":8043},[2587],[507,8045,8047],{"className":8046,"style":6833},[2591],[507,8048],{},[507,8050],{"className":8051,"style":2715},[2714],[507,8053,2691],{"className":8054},[2719],[507,8056],{"className":8057,"style":2715},[2714],[507,8059,8061,8064,8068,8110],{"className":8060},[2561],[507,8062],{"className":8063,"style":7967},[2565],[507,8065,7759],{"className":8066,"style":8067},[2570,2611],"margin-right:0.0962em;",[507,8069,8071,8075],{"className":8070},[2570],[507,8072,7764],{"className":8073,"style":8074},[2570,2611],"margin-right:0.0715em;",[507,8076,8078],{"className":8077},[2579],[507,8079,8081,8102],{"className":8080},[2583,3200],[507,8082,8084,8099],{"className":8083},[2587],[507,8085,8087],{"className":8086,"style":6802},[2591],[507,8088,8090,8093],{"style":8089},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[507,8091],{"className":8092,"style":2600},[2599],[507,8094,8096],{"className":8095},[2604,2605,2606,2607],[507,8097,3293],{"className":8098},[2570,2611,2607],[507,8100,3225],{"className":8101},[3224],[507,8103,8105],{"className":8104},[2587],[507,8106,8108],{"className":8107,"style":3232},[2591],[507,8109],{},[507,8111,8113,8116],{"className":8112},[2570],[507,8114,7764],{"className":8115,"style":8074},[2570,2611],[507,8117,8119],{"className":8118},[2579],[507,8120,8122,8142],{"className":8121},[2583,3200],[507,8123,8125,8139],{"className":8124},[2587],[507,8126,8128],{"className":8127,"style":6802},[2591],[507,8129,8130,8133],{"style":8089},[507,8131],{"className":8132,"style":2600},[2599],[507,8134,8136],{"className":8135},[2604,2605,2606,2607],[507,8137,2372],{"className":8138,"style":6823},[2570,2611,2607],[507,8140,3225],{"className":8141},[3224],[507,8143,8145],{"className":8144},[2587],[507,8146,8148],{"className":8147,"style":6833},[2591],[507,8149],{},[498,8151,8153],{"className":500,"code":8152,"language":502,"meta":104,"style":104},"# Define problem Hamiltonian.\nn_qubits = 30\nJ = 1  # coupling strength for ZZ interaction\n\n# Define the Hamiltonian:\nH_int = [[\"I\"] * n_qubits for _ in range(3 * (n_qubits - 1))]\nfor i in range(n_qubits - 1):\n    H_int[i][i] = \"Z\"\n    H_int[i][i + 1] = \"Z\"\nfor i in range(n_qubits - 1):\n    H_int[n_qubits - 1 + i][i] = \"X\"\n    H_int[n_qubits - 1 + i][i + 1] = \"X\"\nfor i in range(n_qubits - 1):\n    H_int[2 * (n_qubits - 1) + i][i] = \"Y\"\n    H_int[2 * (n_qubits - 1) + i][i + 1] = \"Y\"\nH_int = [\"\".join(term) for term in H_int]\nH_tot = [(term, J) if term.count(\"Z\") == 2 else (term, 1) for term in H_int]\n\n# Get operator\nH_op = SparsePauliOp.from_list(H_tot)\nprint(H_tot)\n",[504,8154,8155,8160,8170,8182,8186,8191,8240,8259,8269,8284,8302,8322,8345,8363,8389,8419,8449,8496,8500,8505,8521],{"__ignoreMap":104},[507,8156,8157],{"class":509,"line":510},[507,8158,8159],{"class":562},"# Define problem Hamiltonian.\n",[507,8161,8162,8165,8167],{"class":509,"line":105},[507,8163,8164],{"class":517},"n_qubits ",[507,8166,573],{"class":572},[507,8168,8169],{"class":583}," 30\n",[507,8171,8172,8175,8177,8179],{"class":509,"line":540},[507,8173,8174],{"class":517},"J ",[507,8176,573],{"class":572},[507,8178,1426],{"class":583},[507,8180,8181],{"class":562},"  # coupling strength for ZZ interaction\n",[507,8183,8184],{"class":509,"line":553},[507,8185,556],{"emptyLinePlaceholder":133},[507,8187,8188],{"class":509,"line":559},[507,8189,8190],{"class":562},"# Define the Hamiltonian:\n",[507,8192,8193,8196,8198,8201,8204,8207,8209,8212,8214,8217,8219,8222,8224,8227,8230,8233,8235,8237],{"class":509,"line":566},[507,8194,8195],{"class":517},"H_int ",[507,8197,573],{"class":572},[507,8199,8200],{"class":517}," [[",[507,8202,8203],{"class":730},"\"I\"",[507,8205,8206],{"class":517},"] ",[507,8208,2391],{"class":572},[507,8210,8211],{"class":517}," n_qubits ",[507,8213,1630],{"class":513},[507,8215,8216],{"class":517}," _ ",[507,8218,1636],{"class":513},[507,8220,8221],{"class":572}," range",[507,8223,580],{"class":517},[507,8225,8226],{"class":583},"3",[507,8228,8229],{"class":572}," *",[507,8231,8232],{"class":517}," (n_qubits ",[507,8234,2367],{"class":572},[507,8236,1426],{"class":583},[507,8238,8239],{"class":517},"))]\n",[507,8241,8242,8244,8247,8249,8251,8253,8255,8257],{"class":509,"line":590},[507,8243,1630],{"class":513},[507,8245,8246],{"class":517}," i ",[507,8248,1636],{"class":513},[507,8250,8221],{"class":572},[507,8252,2104],{"class":517},[507,8254,2367],{"class":572},[507,8256,1426],{"class":583},[507,8258,1883],{"class":517},[507,8260,8261,8264,8266],{"class":509,"line":610},[507,8262,8263],{"class":517},"    H_int[i][i] ",[507,8265,573],{"class":572},[507,8267,8268],{"class":730}," \"Z\"\n",[507,8270,8271,8274,8276,8278,8280,8282],{"class":509,"line":634},[507,8272,8273],{"class":517},"    H_int[i][i ",[507,8275,2107],{"class":572},[507,8277,1426],{"class":583},[507,8279,8206],{"class":517},[507,8281,573],{"class":572},[507,8283,8268],{"class":730},[507,8285,8286,8288,8290,8292,8294,8296,8298,8300],{"class":509,"line":661},[507,8287,1630],{"class":513},[507,8289,8246],{"class":517},[507,8291,1636],{"class":513},[507,8293,8221],{"class":572},[507,8295,2104],{"class":517},[507,8297,2367],{"class":572},[507,8299,1426],{"class":583},[507,8301,1883],{"class":517},[507,8303,8304,8307,8309,8311,8314,8317,8319],{"class":509,"line":678},[507,8305,8306],{"class":517},"    H_int[n_qubits ",[507,8308,2367],{"class":572},[507,8310,1426],{"class":583},[507,8312,8313],{"class":572}," +",[507,8315,8316],{"class":517}," i][i] ",[507,8318,573],{"class":572},[507,8320,8321],{"class":730}," \"X\"\n",[507,8323,8324,8326,8328,8330,8332,8335,8337,8339,8341,8343],{"class":509,"line":683},[507,8325,8306],{"class":517},[507,8327,2367],{"class":572},[507,8329,1426],{"class":583},[507,8331,8313],{"class":572},[507,8333,8334],{"class":517}," i][i ",[507,8336,2107],{"class":572},[507,8338,1426],{"class":583},[507,8340,8206],{"class":517},[507,8342,573],{"class":572},[507,8344,8321],{"class":730},[507,8346,8347,8349,8351,8353,8355,8357,8359,8361],{"class":509,"line":697},[507,8348,1630],{"class":513},[507,8350,8246],{"class":517},[507,8352,1636],{"class":513},[507,8354,8221],{"class":572},[507,8356,2104],{"class":517},[507,8358,2367],{"class":572},[507,8360,1426],{"class":583},[507,8362,1883],{"class":517},[507,8364,8365,8368,8370,8372,8374,8376,8378,8380,8382,8384,8386],{"class":509,"line":710},[507,8366,8367],{"class":517},"    H_int[",[507,8369,584],{"class":583},[507,8371,8229],{"class":572},[507,8373,8232],{"class":517},[507,8375,2367],{"class":572},[507,8377,1426],{"class":583},[507,8379,655],{"class":517},[507,8381,2107],{"class":572},[507,8383,8316],{"class":517},[507,8385,573],{"class":572},[507,8387,8388],{"class":730}," \"Y\"\n",[507,8390,8391,8393,8395,8397,8399,8401,8403,8405,8407,8409,8411,8413,8415,8417],{"class":509,"line":715},[507,8392,8367],{"class":517},[507,8394,584],{"class":583},[507,8396,8229],{"class":572},[507,8398,8232],{"class":517},[507,8400,2367],{"class":572},[507,8402,1426],{"class":583},[507,8404,655],{"class":517},[507,8406,2107],{"class":572},[507,8408,8334],{"class":517},[507,8410,2107],{"class":572},[507,8412,1426],{"class":583},[507,8414,8206],{"class":517},[507,8416,573],{"class":572},[507,8418,8388],{"class":730},[507,8420,8421,8423,8425,8428,8431,8433,8436,8439,8441,8444,8446],{"class":509,"line":721},[507,8422,8195],{"class":517},[507,8424,573],{"class":572},[507,8426,8427],{"class":517}," [",[507,8429,8430],{"class":730},"\"\"",[507,8432,53],{"class":517},[507,8434,8435],{"class":576},"join",[507,8437,8438],{"class":517},"(term) ",[507,8440,1630],{"class":513},[507,8442,8443],{"class":517}," term ",[507,8445,1636],{"class":513},[507,8447,8448],{"class":517}," H_int]\n",[507,8450,8451,8454,8456,8459,8461,8464,8467,8469,8472,8474,8476,8478,8481,8484,8486,8488,8490,8492,8494],{"class":509,"line":736},[507,8452,8453],{"class":517},"H_tot ",[507,8455,573],{"class":572},[507,8457,8458],{"class":517}," [(term, J) ",[507,8460,1645],{"class":513},[507,8462,8463],{"class":517}," term.",[507,8465,8466],{"class":576},"count",[507,8468,580],{"class":517},[507,8470,8471],{"class":730},"\"Z\"",[507,8473,655],{"class":517},[507,8475,1723],{"class":572},[507,8477,2316],{"class":583},[507,8479,8480],{"class":513}," else",[507,8482,8483],{"class":517}," (term, ",[507,8485,625],{"class":583},[507,8487,655],{"class":517},[507,8489,1630],{"class":513},[507,8491,8443],{"class":517},[507,8493,1636],{"class":513},[507,8495,8448],{"class":517},[507,8497,8498],{"class":509,"line":748},[507,8499,556],{"emptyLinePlaceholder":133},[507,8501,8502],{"class":509,"line":761},[507,8503,8504],{"class":562},"# Get operator\n",[507,8506,8507,8510,8512,8515,8518],{"class":509,"line":775},[507,8508,8509],{"class":517},"H_op ",[507,8511,573],{"class":572},[507,8513,8514],{"class":517}," SparsePauliOp.",[507,8516,8517],{"class":576},"from_list",[507,8519,8520],{"class":517},"(H_tot)\n",[507,8522,8523,8526],{"class":509,"line":784},[507,8524,8525],{"class":572},"print",[507,8527,8520],{"class":517},[498,8529,8533],{"className":8530,"code":8532,"language":7039,"meta":104},[8531],"language-text","[('ZZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IZZIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIZZIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIZZIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIZZIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIZZIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIZZIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIZZIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIZZIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIZZIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIZZIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIZZIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIZZIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIZZIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIZZIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIZZIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIZZIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 1), ('XXIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IXXIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIXXIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIXXIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIXXIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIXXIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIXXIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIXXIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIXXIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIXXIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIXXIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIXXIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIXXIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIXXIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIXXIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIXXIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIXXIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIXXIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIXXIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIXXIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIXXIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIXXIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIXXIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIXXIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIXXIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIXXIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIXXII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIXXI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIXX', 1), ('YYIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IYYIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIYYIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIYYIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIYYIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIYYIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIYYIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIYYIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIYYIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIYYIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIYYIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIYYIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIYYIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIYYIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIYYIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIYYIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIYYIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIYYIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIYYIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIYYIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIYYIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIYYIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIYYIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIYYIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIYYIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIYYIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIYYII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIYYI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIYY', 1)]\n",[504,8534,8532],{"__ignoreMap":104},[7660,8536,8538],{"id":8537},"set-parameters-for-the-algorithm","Set parameters for the algorithm",[18,8540,8541,8542,8544,8545,8548,8549,8602,8603,8636],{},"We heuristically choose a value for the time-step ",[504,8543,5463],{}," (based on upper bounds on the Hamiltonian norm). Ref ",[49,8546,8547],{"href":1086},"[2]","  showed that a sufficiently small timestep is ",[507,8550,8552,8579],{"className":8551},[2523],[507,8553,8555],{"className":8554},[2527],[2529,8556,8557],{"xmlns":2531},[2533,8558,8559,8576],{},[2536,8560,8561,8564,8566,8568,8570,8572,8574],{},[2542,8562,8563],{},"π",[2542,8565,645],{"mathvariant":2748},[2542,8567,2749],{"mathvariant":2748},[2542,8569,2749],{"mathvariant":2748},[2542,8571,3138],{},[2542,8573,2749],{"mathvariant":2748},[2542,8575,2749],{"mathvariant":2748},[2549,8577,8578],{"encoding":2551},"\\pi\u002F\\vert \\vert H \\vert \\vert",[507,8580,8582],{"className":8581,"ariaHidden":2557},[2556],[507,8583,8585,8588,8591,8595,8598],{"className":8584},[2561],[507,8586],{"className":8587,"style":2769},[2565],[507,8589,8563],{"className":8590,"style":2776},[2570,2611],[507,8592,8594],{"className":8593},[2570],"\u002F∣∣",[507,8596,3138],{"className":8597,"style":3153},[2570,2611],[507,8599,8601],{"className":8600},[2570],"∣∣",", and that it is preferable up to a point to underestimate this value rather than overestimate, since overestimating can allow contributions from high-energy states to corrupt even the optimal state in the Krylov space. On the other hand, choosing ",[507,8604,8606,8621],{"className":8605},[2523],[507,8607,8609],{"className":8608},[2527],[2529,8610,8611],{"xmlns":2531},[2533,8612,8613,8619],{},[2536,8614,8615,8617],{},[2542,8616,4959],{},[2542,8618,3298],{},[2549,8620,5463],{"encoding":2551},[507,8622,8624],{"className":8623,"ariaHidden":2557},[2556],[507,8625,8627,8630,8633],{"className":8626},[2561],[507,8628],{"className":8629,"style":5434},[2565],[507,8631,4959],{"className":8632},[2570,2611],[507,8634,3298],{"className":8635},[2570,2611]," to be too small leads to worse conditioning of the Krylov subspace, since the Krylov basis vectors differ less from timestep to timestep.",[498,8638,8640],{"className":500,"code":8639,"language":502,"meta":104,"style":104},"# Get Hamiltonian restricted to single-particle states\nsingle_particle_H = np.zeros((n_qubits, n_qubits))\nfor i in range(n_qubits):\n    for j in range(i + 1):\n        for p, coeff in H_op.to_list():\n            p_x = Pauli(p).x\n            p_z = Pauli(p).z\n            if all(\n                p_x[k] == ((i == k) + (j == k)) % 2 for k in range(n_qubits)\n            ):\n                sgn = (\n                    (-1j) ** sum(p_z[k] and p_x[k] for k in range(n_qubits))\n                ) * ((-1) ** p_z[i])\n            else:\n                sgn = 0\n            single_particle_H[i, j] += sgn * coeff\nfor i in range(n_qubits):\n    for j in range(i + 1, n_qubits):\n        single_particle_H[i, j] = np.conj(single_particle_H[j, i])\n\n# Set dt according to spectral norm\ndt = np.pi \u002F np.linalg.norm(single_particle_H, ord=2)\ndt\n",[504,8641,8642,8647,8661,8674,8694,8708,8721,8733,8742,8784,8789,8798,8836,8856,8863,8871,8884,8896,8915,8929,8933,8938,8968],{"__ignoreMap":104},[507,8643,8644],{"class":509,"line":510},[507,8645,8646],{"class":562},"# Get Hamiltonian restricted to single-particle states\n",[507,8648,8649,8652,8654,8656,8658],{"class":509,"line":105},[507,8650,8651],{"class":517},"single_particle_H ",[507,8653,573],{"class":572},[507,8655,1616],{"class":517},[507,8657,2149],{"class":576},[507,8659,8660],{"class":517},"((n_qubits, n_qubits))\n",[507,8662,8663,8665,8667,8669,8671],{"class":509,"line":540},[507,8664,1630],{"class":513},[507,8666,8246],{"class":517},[507,8668,1636],{"class":513},[507,8670,8221],{"class":572},[507,8672,8673],{"class":517},"(n_qubits):\n",[507,8675,8676,8678,8681,8683,8685,8688,8690,8692],{"class":509,"line":553},[507,8677,1916],{"class":513},[507,8679,8680],{"class":517}," j ",[507,8682,1636],{"class":513},[507,8684,8221],{"class":572},[507,8686,8687],{"class":517},"(i ",[507,8689,2107],{"class":572},[507,8691,1426],{"class":583},[507,8693,1883],{"class":517},[507,8695,8696,8698,8700,8702,8704,8706],{"class":509,"line":559},[507,8697,2267],{"class":513},[507,8699,1919],{"class":517},[507,8701,1636],{"class":513},[507,8703,1924],{"class":517},[507,8705,1927],{"class":576},[507,8707,1930],{"class":517},[507,8709,8710,8713,8715,8718],{"class":509,"line":566},[507,8711,8712],{"class":517},"            p_x ",[507,8714,573],{"class":572},[507,8716,8717],{"class":576}," Pauli",[507,8719,8720],{"class":517},"(p).x\n",[507,8722,8723,8726,8728,8730],{"class":509,"line":590},[507,8724,8725],{"class":517},"            p_z ",[507,8727,573],{"class":572},[507,8729,8717],{"class":576},[507,8731,8732],{"class":517},"(p).z\n",[507,8734,8735,8737,8740],{"class":509,"line":610},[507,8736,2298],{"class":513},[507,8738,8739],{"class":572}," all",[507,8741,1376],{"class":517},[507,8743,8744,8747,8749,8752,8754,8757,8759,8762,8764,8767,8770,8772,8775,8777,8779,8781],{"class":509,"line":634},[507,8745,8746],{"class":517},"                p_x[k] ",[507,8748,1723],{"class":572},[507,8750,8751],{"class":517}," ((i ",[507,8753,1723],{"class":572},[507,8755,8756],{"class":517}," k) ",[507,8758,2107],{"class":572},[507,8760,8761],{"class":517}," (j ",[507,8763,1723],{"class":572},[507,8765,8766],{"class":517}," k)) ",[507,8768,8769],{"class":572},"%",[507,8771,2316],{"class":583},[507,8773,8774],{"class":513}," for",[507,8776,1720],{"class":517},[507,8778,1636],{"class":513},[507,8780,8221],{"class":572},[507,8782,8783],{"class":517},"(n_qubits)\n",[507,8785,8786],{"class":509,"line":661},[507,8787,8788],{"class":517},"            ):\n",[507,8790,8791,8794,8796],{"class":509,"line":678},[507,8792,8793],{"class":517},"                sgn ",[507,8795,573],{"class":572},[507,8797,1334],{"class":517},[507,8799,8800,8803,8805,8807,8809,8811,8813,8816,8819,8822,8825,8827,8829,8831,8833],{"class":509,"line":683},[507,8801,8802],{"class":517},"                    (",[507,8804,2367],{"class":572},[507,8806,625],{"class":583},[507,8808,2372],{"class":513},[507,8810,655],{"class":517},[507,8812,2377],{"class":572},[507,8814,8815],{"class":572}," sum",[507,8817,8818],{"class":517},"(p_z[k] ",[507,8820,8821],{"class":513},"and",[507,8823,8824],{"class":517}," p_x[k] ",[507,8826,1630],{"class":513},[507,8828,1720],{"class":517},[507,8830,1636],{"class":513},[507,8832,8221],{"class":572},[507,8834,8835],{"class":517},"(n_qubits))\n",[507,8837,8838,8841,8843,8845,8847,8849,8851,8853],{"class":509,"line":697},[507,8839,8840],{"class":517},"                ) ",[507,8842,2391],{"class":572},[507,8844,2364],{"class":517},[507,8846,2367],{"class":572},[507,8848,625],{"class":583},[507,8850,655],{"class":517},[507,8852,2377],{"class":572},[507,8854,8855],{"class":517}," p_z[i])\n",[507,8857,8858,8861],{"class":509,"line":710},[507,8859,8860],{"class":513},"            else",[507,8862,1728],{"class":517},[507,8864,8865,8867,8869],{"class":509,"line":715},[507,8866,8793],{"class":517},[507,8868,573],{"class":572},[507,8870,2246],{"class":583},[507,8872,8873,8876,8878,8880,8882],{"class":509,"line":721},[507,8874,8875],{"class":517},"            single_particle_H[i, j] ",[507,8877,2285],{"class":572},[507,8879,2455],{"class":517},[507,8881,2391],{"class":572},[507,8883,2460],{"class":517},[507,8885,8886,8888,8890,8892,8894],{"class":509,"line":736},[507,8887,1630],{"class":513},[507,8889,8246],{"class":517},[507,8891,1636],{"class":513},[507,8893,8221],{"class":572},[507,8895,8673],{"class":517},[507,8897,8898,8900,8902,8904,8906,8908,8910,8912],{"class":509,"line":748},[507,8899,1916],{"class":513},[507,8901,8680],{"class":517},[507,8903,1636],{"class":513},[507,8905,8221],{"class":572},[507,8907,8687],{"class":517},[507,8909,2107],{"class":572},[507,8911,1426],{"class":583},[507,8913,8914],{"class":517},", n_qubits):\n",[507,8916,8917,8920,8922,8924,8926],{"class":509,"line":761},[507,8918,8919],{"class":517},"        single_particle_H[i, j] ",[507,8921,573],{"class":572},[507,8923,1616],{"class":517},[507,8925,1674],{"class":576},[507,8927,8928],{"class":517},"(single_particle_H[j, i])\n",[507,8930,8931],{"class":509,"line":775},[507,8932,556],{"emptyLinePlaceholder":133},[507,8934,8935],{"class":509,"line":784},[507,8936,8937],{"class":562},"# Set dt according to spectral norm\n",[507,8939,8940,8943,8945,8948,8950,8953,8956,8959,8962,8964,8966],{"class":509,"line":796},[507,8941,8942],{"class":517},"dt ",[507,8944,573],{"class":572},[507,8946,8947],{"class":517}," np.pi ",[507,8949,645],{"class":572},[507,8951,8952],{"class":517}," np.linalg.",[507,8954,8955],{"class":576},"norm",[507,8957,8958],{"class":517},"(single_particle_H, ",[507,8960,8961],{"class":2155},"ord",[507,8963,573],{"class":572},[507,8965,584],{"class":583},[507,8967,587],{"class":517},[507,8969,8970],{"class":509,"line":809},[507,8971,8972],{"class":517},"dt\n",[498,8974,8977],{"className":8975,"code":8976,"language":7039,"meta":104},[8531],"np.float64(0.10833078115826875)\n",[504,8978,8976],{"__ignoreMap":104},[18,8980,8981],{},"And set other parameters of the algorithm. For the sake of this tutorial, we'll limit ourselves to using a Krylov space with only five dimensions, which is quite limiting.",[498,8983,8985],{"className":500,"code":8984,"language":502,"meta":104,"style":104},"# Set parameters for quantum Krylov algorithm\nkrylov_dim = 5  # size of Krylov subspace\nnum_trotter_steps = 6\ndt_circ = dt \u002F num_trotter_steps\n",[504,8986,8987,8992,9005,9015],{"__ignoreMap":104},[507,8988,8989],{"class":509,"line":510},[507,8990,8991],{"class":562},"# Set parameters for quantum Krylov algorithm\n",[507,8993,8994,8997,8999,9002],{"class":509,"line":105},[507,8995,8996],{"class":517},"krylov_dim ",[507,8998,573],{"class":572},[507,9000,9001],{"class":583}," 5",[507,9003,9004],{"class":562},"  # size of Krylov subspace\n",[507,9006,9007,9010,9012],{"class":509,"line":540},[507,9008,9009],{"class":517},"num_trotter_steps ",[507,9011,573],{"class":572},[507,9013,9014],{"class":583}," 6\n",[507,9016,9017,9020,9022,9025,9027],{"class":509,"line":553},[507,9018,9019],{"class":517},"dt_circ ",[507,9021,573],{"class":572},[507,9023,9024],{"class":517}," dt ",[507,9026,645],{"class":572},[507,9028,9029],{"class":517}," num_trotter_steps\n",[7660,9031,9033],{"id":9032},"state-preparation","State preparation",[18,9035,9036,9037,9076,9077,9115],{},"Pick a reference state ",[507,9038,9040,9058],{"className":9039},[2523],[507,9041,9043],{"className":9042},[2527],[2529,9044,9045],{"xmlns":2531},[2533,9046,9047,9055],{},[2536,9048,9049,9051,9053],{},[2542,9050,2749],{"mathvariant":2748},[2542,9052,3793],{},[2689,9054,2756],{"stretchy":2755},[2549,9056,9057],{"encoding":2551},"\\vert \\psi \\rangle",[507,9059,9061],{"className":9060,"ariaHidden":2557},[2556],[507,9062,9064,9067,9070,9073],{"className":9063},[2561],[507,9065],{"className":9066,"style":2769},[2565],[507,9068,2749],{"className":9069},[2570],[507,9071,3793],{"className":9072,"style":2776},[2570,2611],[507,9074,2756],{"className":9075},[2780]," that has some overlap with the ground state. For this Hamiltonian, We use the a state with an excitation in the middle qubit ",[507,9078,9080,9099],{"className":9079},[2523],[507,9081,9083],{"className":9082},[2527],[2529,9084,9085],{"xmlns":2531},[2533,9086,9087,9096],{},[2536,9088,9089,9091,9094],{},[2542,9090,2749],{"mathvariant":2748},[2693,9092,9093],{},"00..010...00",[2689,9095,2756],{"stretchy":2755},[2549,9097,9098],{"encoding":2551},"\\vert 00..010...00 \\rangle",[507,9100,9102],{"className":9101,"ariaHidden":2557},[2556],[507,9103,9105,9108,9112],{"className":9104},[2561],[507,9106],{"className":9107,"style":2769},[2565],[507,9109,9111],{"className":9110},[2570],"∣00..010...00",[507,9113,2756],{"className":9114},[2780]," as our reference state.",[498,9117,9119],{"className":500,"code":9118,"language":502,"meta":104,"style":104},"qc_state_prep = QuantumCircuit(n_qubits)\nqc_state_prep.x(int(n_qubits \u002F 2) + 1)\nqc_state_prep.draw(\"mpl\", scale=0.5)\n",[504,9120,9121,9132,9158],{"__ignoreMap":104},[507,9122,9123,9126,9128,9130],{"class":509,"line":510},[507,9124,9125],{"class":517},"qc_state_prep ",[507,9127,573],{"class":572},[507,9129,577],{"class":576},[507,9131,8783],{"class":517},[507,9133,9134,9137,9140,9142,9144,9146,9148,9150,9152,9154,9156],{"class":509,"line":105},[507,9135,9136],{"class":517},"qc_state_prep.",[507,9138,9139],{"class":576},"x",[507,9141,580],{"class":517},[507,9143,1420],{"class":572},[507,9145,2104],{"class":517},[507,9147,645],{"class":572},[507,9149,2316],{"class":583},[507,9151,655],{"class":517},[507,9153,2107],{"class":572},[507,9155,1426],{"class":583},[507,9157,587],{"class":517},[507,9159,9160,9162,9165,9167,9170,9172,9175,9177,9180],{"class":509,"line":540},[507,9161,9136],{"class":517},[507,9163,9164],{"class":576},"draw",[507,9166,580],{"class":517},[507,9168,9169],{"class":730},"\"mpl\"",[507,9171,622],{"class":517},[507,9173,9174],{"class":2155},"scale",[507,9176,573],{"class":572},[507,9178,9179],{"class":583},"0.5",[507,9181,587],{"class":517},[831,9183],{"alt":9184,"src":9185},"Output of the previous code cell","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-01.avif",[7660,9187,9189],{"id":9188},"time-evolution","Time evolution",[18,9191,9192,9193,9293,9294,53],{},"We can realize the time-evolution operator generated by a given Hamiltonian: ",[507,9194,9196,9225],{"className":9195},[2523],[507,9197,9199],{"className":9198},[2527],[2529,9200,9201],{"xmlns":2531},[2533,9202,9203,9223],{},[2536,9204,9205,9207,9209],{},[2542,9206,3279],{},[2689,9208,573],{},[2539,9210,9211,9213],{},[2542,9212,3286],{},[2536,9214,9215,9217,9219,9221],{},[2689,9216,2691],{},[2542,9218,3293],{},[2542,9220,3138],{},[2542,9222,3298],{},[2549,9224,3301],{"encoding":2551},[507,9226,9228,9246],{"className":9227,"ariaHidden":2557},[2556],[507,9229,9231,9234,9237,9240,9243],{"className":9230},[2561],[507,9232],{"className":9233,"style":2566},[2565],[507,9235,3279],{"className":9236,"style":3314},[2570,2611],[507,9238],{"className":9239,"style":2919},[2714],[507,9241,573],{"className":9242},[2923],[507,9244],{"className":9245,"style":2919},[2714],[507,9247,9249,9252],{"className":9248},[2561],[507,9250],{"className":9251,"style":3330},[2565],[507,9253,9255,9258],{"className":9254},[2570],[507,9256,3286],{"className":9257},[2570,2611],[507,9259,9261],{"className":9260},[2579],[507,9262,9264],{"className":9263},[2583],[507,9265,9267],{"className":9266},[2587],[507,9268,9270],{"className":9269,"style":3330},[2591],[507,9271,9272,9275],{"style":2595},[507,9273],{"className":9274,"style":2600},[2599],[507,9276,9278],{"className":9277},[2604,2605,2606,2607],[507,9279,9281,9284,9287,9290],{"className":9280},[2570,2607],[507,9282,2691],{"className":9283},[2570,2607],[507,9285,3293],{"className":9286},[2570,2611,2607],[507,9288,3138],{"className":9289,"style":3153},[2570,2611,2607],[507,9291,3298],{"className":9292},[2570,2611,2607]," via the ",[49,9295,9297],{"href":9296},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fqiskit.synthesis.LieTrotter","Lie-Trotter approximation",[498,9299,9301],{"className":500,"code":9300,"language":502,"meta":104,"style":104},"t = Parameter(\"t\")\n\n## Create the time-evo op circuit\nevol_gate = PauliEvolutionGate(\n    H_op, time=t, synthesis=LieTrotter(reps=num_trotter_steps)\n)\n\nqr = QuantumRegister(n_qubits)\nqc_evol = QuantumCircuit(qr)\nqc_evol.append(evol_gate, qargs=qr)\n",[504,9302,9303,9320,9324,9329,9341,9372,9376,9380,9392,9404],{"__ignoreMap":104},[507,9304,9305,9308,9310,9313,9315,9318],{"class":509,"line":510},[507,9306,9307],{"class":517},"t ",[507,9309,573],{"class":572},[507,9311,9312],{"class":576}," Parameter",[507,9314,580],{"class":517},[507,9316,9317],{"class":730},"\"t\"",[507,9319,587],{"class":517},[507,9321,9322],{"class":509,"line":105},[507,9323,556],{"emptyLinePlaceholder":133},[507,9325,9326],{"class":509,"line":540},[507,9327,9328],{"class":562},"## Create the time-evo op circuit\n",[507,9330,9331,9334,9336,9339],{"class":509,"line":553},[507,9332,9333],{"class":517},"evol_gate ",[507,9335,573],{"class":572},[507,9337,9338],{"class":576}," PauliEvolutionGate",[507,9340,1376],{"class":517},[507,9342,9343,9346,9349,9351,9354,9357,9359,9362,9364,9367,9369],{"class":509,"line":559},[507,9344,9345],{"class":517},"    H_op, ",[507,9347,9348],{"class":2155},"time",[507,9350,573],{"class":572},[507,9352,9353],{"class":517},"t, ",[507,9355,9356],{"class":2155},"synthesis",[507,9358,573],{"class":572},[507,9360,9361],{"class":576},"LieTrotter",[507,9363,580],{"class":517},[507,9365,9366],{"class":2155},"reps",[507,9368,573],{"class":572},[507,9370,9371],{"class":517},"num_trotter_steps)\n",[507,9373,9374],{"class":509,"line":566},[507,9375,587],{"class":517},[507,9377,9378],{"class":509,"line":590},[507,9379,556],{"emptyLinePlaceholder":133},[507,9381,9382,9385,9387,9390],{"class":509,"line":610},[507,9383,9384],{"class":517},"qr ",[507,9386,573],{"class":572},[507,9388,9389],{"class":576}," QuantumRegister",[507,9391,8783],{"class":517},[507,9393,9394,9397,9399,9401],{"class":509,"line":634},[507,9395,9396],{"class":517},"qc_evol ",[507,9398,573],{"class":572},[507,9400,577],{"class":576},[507,9402,9403],{"class":517},"(qr)\n",[507,9405,9406,9409,9411,9414,9417,9419],{"class":509,"line":661},[507,9407,9408],{"class":517},"qc_evol.",[507,9410,1939],{"class":576},[507,9412,9413],{"class":517},"(evol_gate, ",[507,9415,9416],{"class":2155},"qargs",[507,9418,573],{"class":572},[507,9420,9421],{"class":517},"qr)\n",[498,9423,9426],{"className":9424,"code":9425,"language":7039,"meta":104},[8531],"\u003Cqiskit.circuit.instructionset.InstructionSet at 0x11eef9be0>\n",[504,9427,9425],{"__ignoreMap":104},[7660,9429,9431],{"id":9430},"hadamard-test","Hadamard 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   |0\\rangle|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle + |1\\rangle \\Big)|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle|0\\rangle^N+|1\\rangle |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |0\\rangle^N+|1\\rangle P |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad\\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |\\psi_j\\rangle+|1\\rangle 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are controlled operations that prepare ",[507,10966,10968,10990],{"className":10967},[2523],[507,10969,10971],{"className":10970},[2527],[2529,10972,10973],{"xmlns":2531},[2533,10974,10975,10987],{},[2536,10976,10977,10979,10985],{},[2542,10978,2749],{"mathvariant":2748},[3168,10980,10981,10983],{},[2542,10982,3793],{},[2542,10984,3293],{},[2689,10986,2756],{"stretchy":2755},[2549,10988,10989],{"encoding":2551},"|\\psi_i\\rangle",[507,10991,10993],{"className":10992,"ariaHidden":2557},[2556],[507,10994,10996,10999,11002,11042],{"className":10995},[2561],[507,10997],{"className":10998,"style":2769},[2565],[507,11000,2749],{"className":11001},[2570],[507,11003,11005,11008],{"className":11004},[2570],[507,11006,3793],{"className":11007,"style":2776},[2570,2611],[507,11009,11011],{"className":11010},[2579],[507,11012,11014,11034],{"className":11013},[2583,3200],[507,11015,11017,11031],{"className":11016},[2587],[507,11018,11020],{"className":11019,"style":6802},[2591],[507,11021,11022,11025],{"style":4572},[507,11023],{"className":11024,"style":2600},[2599],[507,11026,11028],{"className":11027},[2604,2605,2606,2607],[507,11029,3293],{"className":11030},[2570,2611,2607],[507,11032,3225],{"className":11033},[3224],[507,11035,11037],{"className":11036},[2587],[507,11038,11040],{"className":11039,"style":3232},[2591],[507,11041],{},[507,11043,2756],{"className":11044},[2780],[507,11046,11048,11070],{"className":11047},[2523],[507,11049,11051],{"className":11050},[2527],[2529,11052,11053],{"xmlns":2531},[2533,11054,11055,11067],{},[2536,11056,11057,11059,11065],{},[2542,11058,2749],{"mathvariant":2748},[3168,11060,11061,11063],{},[2542,11062,3793],{},[2542,11064,2372],{},[2689,11066,2756],{"stretchy":2755},[2549,11068,11069],{"encoding":2551},"|\\psi_j\\rangle",[507,11071,11073],{"className":11072,"ariaHidden":2557},[2556],[507,11074,11076,11079,11082,11122],{"className":11075},[2561],[507,11077],{"className":11078,"style":6951},[2565],[507,11080,2749],{"className":11081},[2570],[507,11083,11085,11088],{"className":11084},[2570],[507,11086,3793],{"className":11087,"style":2776},[2570,2611],[507,11089,11091],{"className":11090},[2579],[507,11092,11094,11114],{"className":11093},[2583,3200],[507,11095,11097,11111],{"className":11096},[2587],[507,11098,11100],{"className":11099,"style":6802},[2591],[507,11101,11102,11105],{"style":4572},[507,11103],{"className":11104,"style":2600},[2599],[507,11106,11108],{"className":11107},[2604,2605,2606,2607],[507,11109,2372],{"className":11110,"style":6823},[2570,2611,2607],[507,11112,3225],{"className":11113},[3224],[507,11115,11117],{"className":11116},[2587],[507,11118,11120],{"className":11119,"style":6833},[2591],[507,11121],{},[507,11123,2756],{"className":11124},[2780]," vectors of the unitary Krylov space, with ",[507,11127,11129,11213],{"className":11128},[2523],[507,11130,11132],{"className":11131},[2527],[2529,11133,11134],{"xmlns":2531},[2533,11135,11136,11210],{},[2536,11137,11138,11140,11146,11148,11150,11168,11170,11172,11174,11176,11194,11200,11202,11204],{},[2542,11139,2749],{"mathvariant":2748},[3168,11141,11142,11144],{},[2542,11143,3793],{},[2542,11145,3626],{},[2689,11147,2756],{"stretchy":2755},[2689,11149,573],{},[2539,11151,11152,11154],{},[2542,11153,3286],{},[2536,11155,11156,11158,11160,11162,11164,11166],{},[2689,11157,2691],{},[2542,11159,3293],{},[2542,11161,3138],{},[2542,11163,3626],{},[2542,11165,4959],{},[2542,11167,3298],{},[2542,11169,2749],{"mathvariant":2748},[2542,11171,3793],{},[2689,11173,2756],{"stretchy":2755},[2689,11175,573],{},[2539,11177,11178,11180],{},[2542,11179,3286],{},[2536,11181,11182,11184,11186,11188,11190,11192],{},[2689,11183,2691],{},[2542,11185,3293],{},[2542,11187,3138],{},[2542,11189,3626],{},[2542,11191,4959],{},[2542,11193,3298],{},[3168,11195,11196,11198],{},[2542,11197,3279],{},[2542,11199,3793],{},[2542,11201,2749],{"mathvariant":2748},[2693,11203,601],{},[2539,11205,11206,11208],{},[2689,11207,2756],{"stretchy":2755},[2542,11209,7681],{},[2549,11211,11212],{"encoding":2551},"|\\psi_k\\rangle = e^{-i H k dt } \\vert \\psi \\rangle = e^{-i H k dt } U_{\\psi} \\vert 0 \\rangle^N",[507,11214,11216,11278,11350],{"className":11215,"ariaHidden":2557},[2556],[507,11217,11219,11222,11225,11266,11269,11272,11275],{"className":11218},[2561],[507,11220],{"className":11221,"style":2769},[2565],[507,11223,2749],{"className":11224},[2570],[507,11226,11228,11231],{"className":11227},[2570],[507,11229,3793],{"className":11230,"style":2776},[2570,2611],[507,11232,11234],{"className":11233},[2579],[507,11235,11237,11258],{"className":11236},[2583,3200],[507,11238,11240,11255],{"className":11239},[2587],[507,11241,11244],{"className":11242,"style":11243},[2591],"height:0.3361em;",[507,11245,11246,11249],{"style":4572},[507,11247],{"className":11248,"style":2600},[2599],[507,11250,11252],{"className":11251},[2604,2605,2606,2607],[507,11253,3626],{"className":11254,"style":3692},[2570,2611,2607],[507,11256,3225],{"className":11257},[3224],[507,11259,11261],{"className":11260},[2587],[507,11262,11264],{"className":11263,"style":3232},[2591],[507,11265],{},[507,11267,2756],{"className":11268},[2780],[507,11270],{"className":11271,"style":2919},[2714],[507,11273,573],{"className":11274},[2923],[507,11276],{"className":11277,"style":2919},[2714],[507,11279,11281,11285,11332,11335,11338,11341,11344,11347],{"className":11280},[2561],[507,11282],{"className":11283,"style":11284},[2565],"height:1.0991em;vertical-align:-0.25em;",[507,11286,11288,11291],{"className":11287},[2570],[507,11289,3286],{"className":11290},[2570,2611],[507,11292,11294],{"className":11293},[2579],[507,11295,11297],{"className":11296},[2583],[507,11298,11300],{"className":11299},[2587],[507,11301,11303],{"className":11302,"style":3662},[2591],[507,11304,11305,11308],{"style":2595},[507,11306],{"className":11307,"style":2600},[2599],[507,11309,11311],{"className":11310},[2604,2605,2606,2607],[507,11312,11314,11317,11320,11323,11326,11329],{"className":11313},[2570,2607],[507,11315,2691],{"className":11316},[2570,2607],[507,11318,3293],{"className":11319},[2570,2611,2607],[507,11321,3138],{"className":11322,"style":3153},[2570,2611,2607],[507,11324,3626],{"className":11325,"style":3692},[2570,2611,2607],[507,11327,4959],{"className":11328},[2570,2611,2607],[507,11330,3298],{"className":11331},[2570,2611,2607],[507,11333,2749],{"className":11334},[2570],[507,11336,3793],{"className":11337,"style":2776},[2570,2611],[507,11339,2756],{"className":11340},[2780],[507,11342],{"className":11343,"style":2919},[2714],[507,11345,573],{"className":11346},[2923],[507,11348],{"className":11349,"style":2919},[2714],[507,11351,11353,11357,11404,11448,11451],{"className":11352},[2561],[507,11354],{"className":11355,"style":11356},[2565],"height:1.1352em;vertical-align:-0.2861em;",[507,11358,11360,11363],{"className":11359},[2570],[507,11361,3286],{"className":11362},[2570,2611],[507,11364,11366],{"className":11365},[2579],[507,11367,11369],{"className":11368},[2583],[507,11370,11372],{"className":11371},[2587],[507,11373,11375],{"className":11374,"style":3662},[2591],[507,11376,11377,11380],{"style":2595},[507,11378],{"className":11379,"style":2600},[2599],[507,11381,11383],{"className":11382},[2604,2605,2606,2607],[507,11384,11386,11389,11392,11395,11398,11401],{"className":11385},[2570,2607],[507,11387,2691],{"className":11388},[2570,2607],[507,11390,3293],{"className":11391},[2570,2611,2607],[507,11393,3138],{"className":11394,"style":3153},[2570,2611,2607],[507,11396,3626],{"className":11397,"style":3692},[2570,2611,2607],[507,11399,4959],{"className":11400},[2570,2611,2607],[507,11402,3298],{"className":11403},[2570,2611,2607],[507,11405,11407,11410],{"className":11406},[2570],[507,11408,3279],{"className":11409,"style":3314},[2570,2611],[507,11411,11413],{"className":11412},[2579],[507,11414,11416,11440],{"className":11415},[2583,3200],[507,11417,11419,11437],{"className":11418},[2587],[507,11420,11422],{"className":11421,"style":11243},[2591],[507,11423,11425,11428],{"style":11424},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[507,11426],{"className":11427,"style":2600},[2599],[507,11429,11431],{"className":11430},[2604,2605,2606,2607],[507,11432,11434],{"className":11433},[2570,2607],[507,11435,3793],{"className":11436,"style":2776},[2570,2611,2607],[507,11438,3225],{"className":11439},[3224],[507,11441,11443],{"className":11442},[2587],[507,11444,11446],{"className":11445,"style":6833},[2591],[507,11447],{},[507,11449,9735],{"className":11450},[2570],[507,11452,11454,11457],{"className":11453},[2780],[507,11455,2756],{"className":11456},[2780],[507,11458,11460],{"className":11459},[2579],[507,11461,11463],{"className":11462},[2583],[507,11464,11466],{"className":11465},[2587],[507,11467,11469],{"className":11468,"style":3330},[2591],[507,11470,11471,11474],{"style":2595},[507,11472],{"className":11473,"style":2600},[2599],[507,11475,11477],{"className":11476},[2604,2605,2606,2607],[507,11478,7681],{"className":11479,"style":3314},[2570,2611,2607],". To measure ",[507,11482,11484,11497],{"className":11483},[2523],[507,11485,11487],{"className":11486},[2527],[2529,11488,11489],{"xmlns":2531},[2533,11490,11491,11495],{},[2536,11492,11493],{},[2542,11494,7731],{},[2549,11496,7731],{"encoding":2551},[507,11498,11500],{"className":11499,"ariaHidden":2557},[2556],[507,11501,11503,11506],{"className":11502},[2561],[507,11504],{"className":11505,"style":2566},[2565],[507,11507,7731],{"className":11508,"style":7876},[2570,2611],", first apply ",[507,11511,11513,11526],{"className":11512},[2523],[507,11514,11516],{"className":11515},[2527],[2529,11517,11518],{"xmlns":2531},[2533,11519,11520,11524],{},[2536,11521,11522],{},[2542,11523,3138],{},[2549,11525,3138],{"encoding":2551},[507,11527,11529],{"className":11528,"ariaHidden":2557},[2556],[507,11530,11532,11535],{"className":11531},[2561],[507,11533],{"className":11534,"style":2566},[2565],[507,11536,3138],{"className":11537,"style":3153},[2570,2611],[507,11539,11541],{"className":11540},[2784],[507,11542,11544,11650],{"className":11543},[2523],[507,11545,11547],{"className":11546},[2527],[2529,11548,11549],{"xmlns":2531,"display":2793},[2533,11550,11551,11647],{},[9452,11552,11553],{"rowspacing":9454,"columnspacing":9455},[9457,11554,11555],{},[9460,11556,11557],{},[9463,11558,11559],{"scriptlevel":601,"displaystyle":2557},[2536,11560,11561,11563,11565,11571,11573,11575,11577,11579,11581,11587,11589,11591,11593,11595,11601,11603,11605,11607,11613,11615,11617,11619,11621,11623,11629,11631,11633,11635,11637,11643,11645],{},[2689,11562,9487],{},[2714,11564],{"width":9455},[9491,11566,11567,11569],{},[2693,11568,625],{},[2693,11570,584],{},[2542,11572,2749],{"mathvariant":2748},[2693,11574,601],{},[2689,11576,2756],{"stretchy":2755},[2689,11578,580],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2542,11580,2749],{"mathvariant":2748},[3168,11582,11583,11585],{},[2542,11584,3793],{},[2542,11586,2372],{},[2689,11588,2756],{"stretchy":2755},[2689,11590,2107],{},[2542,11592,3174],{},[2542,11594,2749],{"mathvariant":2748},[3168,11596,11597,11599],{},[2542,11598,3793],{},[2542,11600,3293],{},[2689,11602,2756],{"stretchy":2755},[2689,11604,3649],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2689,11606,2107],{},[9491,11608,11609,11611],{},[2693,11610,625],{},[2693,11612,584],{},[2542,11614,2749],{"mathvariant":2748},[2693,11616,625],{},[2689,11618,2756],{"stretchy":2755},[2689,11620,580],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2542,11622,2749],{"mathvariant":2748},[3168,11624,11625,11627],{},[2542,11626,3793],{},[2542,11628,2372],{},[2689,11630,2756],{"stretchy":2755},[2689,11632,2691],{},[2542,11634,3174],{},[2542,11636,2749],{"mathvariant":2748},[3168,11638,11639,11641],{},[2542,11640,3793],{},[2542,11642,3293],{},[2689,11644,2756],{"stretchy":2755},[2689,11646,3649],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2549,11648,11649],{"encoding":2551},"\\begin{equation*}\n    \\longrightarrow\\quad\\frac{1}{2}|0\\rangle\\Big( |\\psi_j\\rangle + P|\\psi_i\\rangle\\Big) + \\frac{1}{2}|1\\rangle\\Big(|\\psi_j\\rangle - P|\\psi_i\\rangle\\Big)\n\\end{equation*}",[507,11651,11653],{"className":11652,"ariaHidden":2557},[2556],[507,11654,11656,11660],{"className":11655},[2561],[507,11657],{"className":11658,"style":11659},[2565],"height:2.0074em;vertical-align:-0.7537em;",[507,11661,11663],{"className":11662},[2570],[507,11664,11666],{"className":11665},[9452],[507,11667,11669],{"className":11668},[9711],[507,11670,11672,12079],{"className":11671},[2583,3200],[507,11673,11675,12076],{"className":11674},[2587],[507,11676,11679],{"className":11677,"style":11678},[2591],"height:1.2537em;",[507,11680,11682,11685],{"style":11681},"top:-3.2537em;",[507,11683],{"className":11684,"style":9728},[2599],[507,11686,11688,11691,11694,11697,11761,11764,11767,11773,11776,11816,11819,11822,11825,11828,11831,11834,11874,11877,11883,11886,11889,11892,11954,11957,11960,11966,11969,12009,12012,12015,12018,12021,12024,12027,12067,12070],{"className":11687},[2570],[507,11689,9487],{"className":11690},[2923],[507,11692],{"className":11693,"style":9774},[2714],[507,11695],{"className":11696,"style":2919},[2714],[507,11698,11700,11703,11758],{"className":11699},[2570],[507,11701],{"className":11702},[2941,9793],[507,11704,11706],{"className":11705},[9491],[507,11707,11709,11749],{"className":11708},[2583,3200],[507,11710,11712,11746],{"className":11711},[2587],[507,11713,11715,11727,11735],{"className":11714,"style":9806},[2591],[507,11716,11718,11721],{"style":11717},"top:-2.314em;",[507,11719],{"className":11720,"style":4310},[2599],[507,11722,11724],{"className":11723},[2570],[507,11725,584],{"className":11726},[2570],[507,11728,11729,11732],{"style":9878},[507,11730],{"className":11731,"style":4310},[2599],[507,11733],{"className":11734,"style":9886},[9885],[507,11736,11737,11740],{"style":9889},[507,11738],{"className":11739,"style":4310},[2599],[507,11741,11743],{"className":11742},[2570],[507,11744,625],{"className":11745},[2570],[507,11747,3225],{"className":11748},[3224],[507,11750,11752],{"className":11751},[2587],[507,11753,11756],{"className":11754,"style":11755},[2591],"height:0.686em;",[507,11757],{},[507,11759],{"className":11760},[2780,9793],[507,11762,9735],{"className":11763},[2570],[507,11765,2756],{"className":11766},[2780],[507,11768,11770],{"className":11769},[2570],[507,11771,580],{"className":11772},[2947,9920],[507,11774,2749],{"className":11775},[2570],[507,11777,11779,11782],{"className":11778},[2570],[507,11780,3793],{"className":11781,"style":2776},[2570,2611],[507,11783,11785],{"className":11784},[2579],[507,11786,11788,11808],{"className":11787},[2583,3200],[507,11789,11791,11805],{"className":11790},[2587],[507,11792,11794],{"className":11793,"style":6802},[2591],[507,11795,11796,11799],{"style":4572},[507,11797],{"className":11798,"style":2600},[2599],[507,11800,11802],{"className":11801},[2604,2605,2606,2607],[507,11803,2372],{"className":11804,"style":6823},[2570,2611,2607],[507,11806,3225],{"className":11807},[3224],[507,11809,11811],{"className":11810},[2587],[507,11812,11814],{"className":11813,"style":6833},[2591],[507,11815],{},[507,11817,2756],{"className":11818},[2780],[507,11820],{"className":11821,"style":2715},[2714],[507,11823,2107],{"className":11824},[2719],[507,11826],{"className":11827,"style":2715},[2714],[507,11829,3174],{"className":11830,"style":3220},[2570,2611],[507,11832,2749],{"className":11833},[2570],[507,11835,11837,11840],{"className":11836},[2570],[507,11838,3793],{"className":11839,"style":2776},[2570,2611],[507,11841,11843],{"className":11842},[2579],[507,11844,11846,11866],{"className":11845},[2583,3200],[507,11847,11849,11863],{"className":11848},[2587],[507,11850,11852],{"className":11851,"style":6802},[2591],[507,11853,11854,11857],{"style":4572},[507,11855],{"className":11856,"style":2600},[2599],[507,11858,11860],{"className":11859},[2604,2605,2606,2607],[507,11861,3293],{"className":11862},[2570,2611,2607],[507,11864,3225],{"className":11865},[3224],[507,11867,11869],{"className":11868},[2587],[507,11870,11872],{"className":11871,"style":3232},[2591],[507,11873],{},[507,11875,2756],{"className":11876},[2780],[507,11878,11880],{"className":11879},[2570],[507,11881,3649],{"className":11882},[2947,9920],[507,11884],{"className":11885,"style":2715},[2714],[507,11887,2107],{"className":11888},[2719],[507,11890],{"className":11891,"style":2715},[2714],[507,11893,11895,11898,11951],{"className":11894},[2570],[507,11896],{"className":11897},[2941,9793],[507,11899,11901],{"className":11900},[9491],[507,11902,11904,11943],{"className":11903},[2583,3200],[507,11905,11907,11940],{"className":11906},[2587],[507,11908,11910,11921,11929],{"className":11909,"style":9806},[2591],[507,11911,11912,11915],{"style":11717},[507,11913],{"className":11914,"style":4310},[2599],[507,11916,11918],{"className":11917},[2570],[507,11919,584],{"className":11920},[2570],[507,11922,11923,11926],{"style":9878},[507,11924],{"className":11925,"style":4310},[2599],[507,11927],{"className":11928,"style":9886},[9885],[507,11930,11931,11934],{"style":9889},[507,11932],{"className":11933,"style":4310},[2599],[507,11935,11937],{"className":11936},[2570],[507,11938,625],{"className":11939},[2570],[507,11941,3225],{"className":11942},[3224],[507,11944,11946],{"className":11945},[2587],[507,11947,11949],{"className":11948,"style":11755},[2591],[507,11950],{},[507,11952],{"className":11953},[2780,9793],[507,11955,9939],{"className":11956},[2570],[507,11958,2756],{"className":11959},[2780],[507,11961,11963],{"className":11962},[2570],[507,11964,580],{"className":11965},[2947,9920],[507,11967,2749],{"className":11968},[2570],[507,11970,11972,11975],{"className":11971},[2570],[507,11973,3793],{"className":11974,"style":2776},[2570,2611],[507,11976,11978],{"className":11977},[2579],[507,11979,11981,12001],{"className":11980},[2583,3200],[507,11982,11984,11998],{"className":11983},[2587],[507,11985,11987],{"className":11986,"style":6802},[2591],[507,11988,11989,11992],{"style":4572},[507,11990],{"className":11991,"style":2600},[2599],[507,11993,11995],{"className":11994},[2604,2605,2606,2607],[507,11996,2372],{"className":11997,"style":6823},[2570,2611,2607],[507,11999,3225],{"className":12000},[3224],[507,12002,12004],{"className":12003},[2587],[507,12005,12007],{"className":12006,"style":6833},[2591],[507,12008],{},[507,12010,2756],{"className":12011},[2780],[507,12013],{"className":12014,"style":2715},[2714],[507,12016,2691],{"className":12017},[2719],[507,12019],{"className":12020,"style":2715},[2714],[507,12022,3174],{"className":12023,"style":3220},[2570,2611],[507,12025,2749],{"className":12026},[2570],[507,12028,12030,12033],{"className":12029},[2570],[507,12031,3793],{"className":12032,"style":2776},[2570,2611],[507,12034,12036],{"className":12035},[2579],[507,12037,12039,12059],{"className":12038},[2583,3200],[507,12040,12042,12056],{"className":12041},[2587],[507,12043,12045],{"className":12044,"style":6802},[2591],[507,12046,12047,12050],{"style":4572},[507,12048],{"className":12049,"style":2600},[2599],[507,12051,12053],{"className":12052},[2604,2605,2606,2607],[507,12054,3293],{"className":12055},[2570,2611,2607],[507,12057,3225],{"className":12058},[3224],[507,12060,12062],{"className":12061},[2587],[507,12063,12065],{"className":12064,"style":3232},[2591],[507,12066],{},[507,12068,2756],{"className":12069},[2780],[507,12071,12073],{"className":12072},[2570],[507,12074,3649],{"className":12075},[2947,9920],[507,12077,3225],{"className":12078},[3224],[507,12080,12082],{"className":12081},[2587],[507,12083,12086],{"className":12084,"style":12085},[2591],"height:0.7537em;",[507,12087],{},[18,12089,12090],{},"... then measure:",[507,12092,12094],{"className":12093},[2784],[507,12095,12097,12279],{"className":12096},[2523],[507,12098,12100],{"className":12099},[2527],[2529,12101,12102],{"xmlns":2531,"display":2793},[2533,12103,12104,12276],{},[9452,12105,12106],{"rowspacing":9454,"columnspacing":9455},[9457,12107,12108],{},[9460,12109,12110],{},[9463,12111,12112],{"scriptlevel":601,"displaystyle":2557},[9452,12113,12117,12225],{"rowspacing":12114,"columnalign":12115,"columnspacing":12116},"0.25em","right left","0em",[9457,12118,12119,12136],{},[9460,12120,12121],{},[9463,12122,12123],{"scriptlevel":601,"displaystyle":2557},[2536,12124,12125,12128,12130,12132,12134],{},[2689,12126,12127],{},"⇒",[2714,12129],{"width":9455},[2689,12131,4425],{"stretchy":2755},[2542,12133,7731],{},[2689,12135,2756],{"stretchy":2755},[9460,12137,12138],{},[9463,12139,12140],{"scriptlevel":601,"displaystyle":2557},[2536,12141,12142,12144,12146,12153,12156,12159,12161,12167,12169,12171,12173,12175,12181,12183,12189,12191,12193,12195,12201,12203,12205,12207,12209,12215,12217,12223],{},[2536,12143],{},[2689,12145,573],{},[9491,12147,12148,12150],{},[2693,12149,625],{},[2693,12151,12152],{},"4",[2689,12154,580],{"fence":2755,"stretchy":2557,"minsize":12155,"maxsize":12155},"3em",[2689,12157,12158],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},"∥",[2542,12160,2749],{"mathvariant":2748},[3168,12162,12163,12165],{},[2542,12164,3793],{},[2542,12166,2372],{},[2689,12168,2756],{"stretchy":2755},[2689,12170,2107],{},[2542,12172,3174],{},[2542,12174,2749],{"mathvariant":2748},[3168,12176,12177,12179],{},[2542,12178,3793],{},[2542,12180,3293],{},[2689,12182,2756],{"stretchy":2755},[2539,12184,12185,12187],{},[2689,12186,12158],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2693,12188,584],{},[2689,12190,2691],{},[2689,12192,12158],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2542,12194,2749],{"mathvariant":2748},[3168,12196,12197,12199],{},[2542,12198,3793],{},[2542,12200,2372],{},[2689,12202,2756],{"stretchy":2755},[2689,12204,2691],{},[2542,12206,3174],{},[2542,12208,2749],{"mathvariant":2748},[3168,12210,12211,12213],{},[2542,12212,3793],{},[2542,12214,3293],{},[2689,12216,2756],{"stretchy":2755},[2539,12218,12219,12221],{},[2689,12220,12158],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2693,12222,584],{},[2689,12224,3649],{"fence":2755,"stretchy":2557,"minsize":12155,"maxsize":12155},[9457,12226,12227,12233],{},[9460,12228,12229],{},[9463,12230,12231],{"scriptlevel":601,"displaystyle":2557},[2536,12232],{},[9460,12234,12235],{},[9463,12236,12237],{"scriptlevel":601,"displaystyle":2557},[2536,12238,12239,12241,12243,12246,12249,12251,12257,12259,12261,12263,12269,12271,12274],{},[2536,12240],{},[2689,12242,573],{},[6167,12244,12245],{},"Re",[2689,12247,12248],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},"[",[2689,12250,4425],{"stretchy":2755},[3168,12252,12253,12255],{},[2542,12254,3793],{},[2542,12256,2372],{},[2542,12258,2749],{"mathvariant":2748},[2542,12260,3174],{},[2542,12262,2749],{"mathvariant":2748},[3168,12264,12265,12267],{},[2542,12266,3793],{},[2542,12268,3293],{},[2689,12270,2756],{"stretchy":2755},[2689,12272,12273],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},"]",[2542,12275,53],{"mathvariant":2748},[2549,12277,12278],{"encoding":2551},"\\begin{equation*}\n\\begin{split}\n    \\Rightarrow\\quad\\langle X\\rangle &= \\frac{1}{4}\\Bigg(\\Big\\|| \\psi_j\\rangle + P|\\psi_i\\rangle \\Big\\|^2-\\Big\\||\\psi_j\\rangle - P|\\psi_i\\rangle\\Big\\|^2\\Bigg) \\\\\n    &= \\text{Re}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{split}\n\\end{equation*}",[507,12280,12282],{"className":12281,"ariaHidden":2557},[2556],[507,12283,12285,12289],{"className":12284},[2561],[507,12286],{"className":12287,"style":12288},[2565],"height:5.1em;vertical-align:-2.3em;",[507,12290,12292],{"className":12291},[2570],[507,12293,12295],{"className":12294},[9452],[507,12296,12298],{"className":12297},[9711],[507,12299,12301,13071],{"className":12300},[2583,3200],[507,12302,12304,13068],{"className":12303},[2587],[507,12305,12308],{"className":12306,"style":12307},[2591],"height:2.8em;",[507,12309,12311,12315],{"style":12310},"top:-4.8em;",[507,12312],{"className":12313,"style":12314},[2599],"height:4.8em;",[507,12316,12318],{"className":12317},[2570],[507,12319,12321],{"className":12320},[2570],[507,12322,12324,12385],{"className":12323},[9452],[507,12325,12328],{"className":12326},[12327],"col-align-r",[507,12329,12331,12376],{"className":12330},[2583,3200],[507,12332,12334,12373],{"className":12333},[2587],[507,12335,12337,12364],{"className":12336,"style":12307},[2591],[507,12338,12339,12343],{"style":12310},[507,12340],{"className":12341,"style":12342},[2599],"height:3.75em;",[507,12344,12346,12349,12352,12355,12358,12361],{"className":12345},[2570],[507,12347,12127],{"className":12348},[2923],[507,12350],{"className":12351,"style":9774},[2714],[507,12353],{"className":12354,"style":2919},[2714],[507,12356,4425],{"className":12357},[2941],[507,12359,7731],{"className":12360,"style":7876},[2570,2611],[507,12362,2756],{"className":12363},[2780],[507,12365,12367,12370],{"style":12366},"top:-2.1em;",[507,12368],{"className":12369,"style":12342},[2599],[507,12371],{"className":12372},[2570],[507,12374,3225],{"className":12375},[3224],[507,12377,12379],{"className":12378},[2587],[507,12380,12383],{"className":12381,"style":12382},[2591],"height:2.3em;",[507,12384],{},[507,12386,12389],{"className":12387},[12388],"col-align-l",[507,12390,12392,13060],{"className":12391},[2583,3200],[507,12393,12395,13057],{"className":12394},[2587],[507,12396,12398,12921],{"className":12397,"style":12307},[2591],[507,12399,12400,12403],{"style":12310},[507,12401],{"className":12402,"style":12342},[2599],[507,12404,12406,12409,12412,12415,12418,12480,12487,12533,12536,12576,12579,12582,12585,12588,12591,12594,12634,12637,12702,12705,12708,12711,12748,12751,12791,12794,12797,12800,12803,12806,12809,12849,12852,12915],{"className":12405},[2570],[507,12407],{"className":12408},[2570],[507,12410],{"className":12411,"style":2919},[2714],[507,12413,573],{"className":12414},[2923],[507,12416],{"className":12417,"style":2919},[2714],[507,12419,12421,12424,12477],{"className":12420},[2570],[507,12422],{"className":12423},[2941,9793],[507,12425,12427],{"className":12426},[9491],[507,12428,12430,12469],{"className":12429},[2583,3200],[507,12431,12433,12466],{"className":12432},[2587],[507,12434,12436,12447,12455],{"className":12435,"style":9806},[2591],[507,12437,12438,12441],{"style":11717},[507,12439],{"className":12440,"style":4310},[2599],[507,12442,12444],{"className":12443},[2570],[507,12445,12152],{"className":12446},[2570],[507,12448,12449,12452],{"style":9878},[507,12450],{"className":12451,"style":4310},[2599],[507,12453],{"className":12454,"style":9886},[9885],[507,12456,12457,12460],{"style":9889},[507,12458],{"className":12459,"style":4310},[2599],[507,12461,12463],{"className":12462},[2570],[507,12464,625],{"className":12465},[2570],[507,12467,3225],{"className":12468},[3224],[507,12470,12472],{"className":12471},[2587],[507,12473,12475],{"className":12474,"style":11755},[2591],[507,12476],{},[507,12478],{"className":12479},[2780,9793],[507,12481,12483],{"className":12482},[2570],[507,12484,580],{"className":12485},[2947,12486],"size4",[507,12488,12490],{"className":12489},[2570],[507,12491,12494],{"className":12492},[2947,12493],"mult",[507,12495,12497,12524],{"className":12496},[2583,3200],[507,12498,12500,12521],{"className":12499},[2587],[507,12501,12504],{"className":12502,"style":12503},[2591],"height:1.15em;",[507,12505,12507,12511],{"style":12506},"top:-3.15em;",[507,12508],{"className":12509,"style":12510},[2599],"height:3.8em;",[507,12512,12514],{"style":12513},"width:0.556em;height:1.8em;",[6281,12515,12518],{"xmlns":6283,"width":12516,"height":9502,"viewBox":12517},"0.556em","0 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",[507,13083,13085,13172],{"className":13084},[2523],[507,13086,13088],{"className":13087},[2527],[2529,13089,13090],{"xmlns":2531},[2533,13091,13092,13169],{},[2536,13093,13094,13096,13098,13100,13103,13109,13111,13113,13115,13117,13119,13121,13123,13125,13127,13129,13131,13133,13135,13141,13143,13145,13147,13153,13155,13157,13159,13161,13163,13165,13167],{},[2542,13095,2749],{"mathvariant":2748},[2542,13097,49],{},[2689,13099,2107],{},[2542,13101,13102],{},"b",[2539,13104,13105,13107],{},[2542,13106,12158],{"mathvariant":2748},[2693,13108,584],{},[2689,13110,573],{},[2689,13112,4425],{"stretchy":2755},[2542,13114,49],{},[2689,13116,2107],{},[2542,13118,13102],{},[2542,13120,2749],{"mathvariant":2748},[2542,13122,49],{},[2689,13124,2107],{},[2542,13126,13102],{},[2689,13128,2756],{"stretchy":2755},[2689,13130,573],{},[2542,13132,12158],{"mathvariant":2748},[2542,13134,49],{},[2539,13136,13137,13139],{},[2542,13138,12158],{"mathvariant":2748},[2693,13140,584],{},[2689,13142,2107],{},[2542,13144,12158],{"mathvariant":2748},[2542,13146,13102],{},[2539,13148,13149,13151],{},[2542,13150,12158],{"mathvariant":2748},[2693,13152,584],{},[2689,13154,2107],{},[2693,13156,584],{},[6167,13158,12245],{},[2689,13160,4425],{"stretchy":2755},[2542,13162,49],{},[2542,13164,2749],{"mathvariant":2748},[2542,13166,13102],{},[2689,13168,2756],{"stretchy":2755},[2549,13170,13171],{"encoding":2551},"|a + b\\|^2 = \\langle a + b | a + b \\rangle = \\|a\\|^2 + \\|b\\|^2 + 2\\text{Re}\\langle a | b \\rangle",[507,13173,13175,13196,13245,13266,13290,13311,13361,13411],{"className":13174,"ariaHidden":2557},[2556],[507,13176,13178,13181,13184,13187,13190,13193],{"className":13177},[2561],[507,13179],{"className":13180,"style":2769},[2565],[507,13182,2749],{"className":13183},[2570],[507,13185,49],{"className":13186},[2570,2611],[507,13188],{"className":13189,"style":2715},[2714],[507,13191,2107],{"className":13192},[2719],[507,13194],{"className":13195,"style":2715},[2714],[507,13197,13199,13203,13206,13236,13239,13242],{"className":13198},[2561],[507,13200],{"className":13201,"style":13202},[2565],"height:1.0641em;vertical-align:-0.25em;",[507,13204,13102],{"className":13205},[2570,2611],[507,13207,13209,13212],{"className":13208},[2570],[507,13210,12158],{"className":13211},[2570],[507,13213,13215],{"className":13214},[2579],[507,13216,13218],{"className":13217},[2583],[507,13219,13221],{"className":13220},[2587],[507,13222,13225],{"className":13223,"style":13224},[2591],"height:0.8141em;",[507,13226,13227,13230],{"style":2595},[507,13228],{"className":13229,"style":2600},[2599],[507,13231,13233],{"className":13232},[2604,2605,2606,2607],[507,13234,584],{"className":13235},[2570,2607],[507,13237],{"className":13238,"style":2919},[2714],[507,13240,573],{"className":13241},[2923],[507,13243],{"className":13244,"style":2919},[2714],[507,13246,13248,13251,13254,13257,13260,13263],{"className":13247},[2561],[507,13249],{"className":13250,"style":2769},[2565],[507,13252,4425],{"className":13253},[2941],[507,13255,49],{"className":13256},[2570,2611],[507,13258],{"className":13259,"style":2715},[2714],[507,13261,2107],{"className":13262},[2719],[507,13264],{"className":13265,"style":2715},[2714],[507,13267,13269,13272,13275,13278,13281,13284,13287],{"className":13268},[2561],[507,13270],{"className":13271,"style":2769},[2565],[507,13273,13102],{"className":13274},[2570,2611],[507,13276,2749],{"className":13277},[2570],[507,13279,49],{"className":13280},[2570,2611],[507,13282],{"className":13283,"style":2715},[2714],[507,13285,2107],{"className":13286},[2719],[507,13288],{"className":13289,"style":2715},[2714],[507,13291,13293,13296,13299,13302,13305,13308],{"className":13292},[2561],[507,13294],{"className":13295,"style":2769},[2565],[507,13297,13102],{"className":13298},[2570,2611],[507,13300,2756],{"className":13301},[2780],[507,13303],{"className":13304,"style":2919},[2714],[507,13306,573],{"className":13307},[2923],[507,13309],{"className":13310,"style":2919},[2714],[507,13312,13314,13317,13320,13323,13352,13355,13358],{"className":13313},[2561],[507,13315],{"className":13316,"style":13202},[2565],[507,13318,12158],{"className":13319},[2570],[507,13321,49],{"className":13322},[2570,2611],[507,13324,13326,13329],{"className":13325},[2570],[507,13327,12158],{"className":13328},[2570],[507,13330,13332],{"className":13331},[2579],[507,13333,13335],{"className":13334},[2583],[507,13336,13338],{"className":13337},[2587],[507,13339,13341],{"className":13340,"style":13224},[2591],[507,13342,13343,13346],{"style":2595},[507,13344],{"className":13345,"style":2600},[2599],[507,13347,13349],{"className":13348},[2604,2605,2606,2607],[507,13350,584],{"className":13351},[2570,2607],[507,13353],{"className":13354,"style":2715},[2714],[507,13356,2107],{"className":13357},[2719],[507,13359],{"className":13360,"style":2715},[2714],[507,13362,13364,13367,13370,13373,13402,13405,13408],{"className":13363},[2561],[507,13365],{"className":13366,"style":13202},[2565],[507,13368,12158],{"className":13369},[2570],[507,13371,13102],{"className":13372},[2570,2611],[507,13374,13376,13379],{"className":13375},[2570],[507,13377,12158],{"className":13378},[2570],[507,13380,13382],{"className":13381},[2579],[507,13383,13385],{"className":13384},[2583],[507,13386,13388],{"className":13387},[2587],[507,13389,13391],{"className":13390,"style":13224},[2591],[507,13392,13393,13396],{"style":2595},[507,13394],{"className":13395,"style":2600},[2599],[507,13397,13399],{"className":13398},[2604,2605,2606,2607],[507,13400,584],{"className":13401},[2570,2607],[507,13403],{"className":13404,"style":2715},[2714],[507,13406,2107],{"className":13407},[2719],[507,13409],{"className":13410,"style":2715},[2714],[507,13412,13414,13417,13420,13426,13429,13432,13435,13438],{"className":13413},[2561],[507,13415],{"className":13416,"style":2769},[2565],[507,13418,584],{"className":13419},[2570],[507,13421,13423],{"className":13422},[2570,7039],[507,13424,12245],{"className":13425},[2570],[507,13427,4425],{"className":13428},[2941],[507,13430,49],{"className":13431},[2570,2611],[507,13433,2749],{"className":13434},[2570],[507,13436,13102],{"className":13437},[2570,2611],[507,13439,2756],{"className":13440},[2780],". Similarly, measuring ",[507,13443,13445,13458],{"className":13444},[2523],[507,13446,13448],{"className":13447},[2527],[2529,13449,13450],{"xmlns":2531},[2533,13451,13452,13456],{},[2536,13453,13454],{},[2542,13455,7746],{},[2549,13457,7746],{"encoding":2551},[507,13459,13461],{"className":13460,"ariaHidden":2557},[2556],[507,13462,13464,13467],{"className":13463},[2561],[507,13465],{"className":13466,"style":2566},[2565],[507,13468,7746],{"className":13469,"style":2715},[2570,2611]," yields",[507,13472,13474],{"className":13473},[2784],[507,13475,13477,13536],{"className":13476},[2523],[507,13478,13480],{"className":13479},[2527],[2529,13481,13482],{"xmlns":2531,"display":2793},[2533,13483,13484,13533],{},[9452,13485,13486],{"rowspacing":9454,"columnspacing":9455},[9457,13487,13488],{},[9460,13489,13490],{},[9463,13491,13492],{"scriptlevel":601,"displaystyle":2557},[2536,13493,13494,13496,13498,13500,13502,13505,13507,13509,13515,13517,13519,13521,13527,13529,13531],{},[2689,13495,4425],{"stretchy":2755},[2542,13497,7746],{},[2689,13499,2756],{"stretchy":2755},[2689,13501,573],{},[6167,13503,13504],{},"Im",[2689,13506,12248],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2689,13508,4425],{"stretchy":2755},[3168,13510,13511,13513],{},[2542,13512,3793],{},[2542,13514,2372],{},[2542,13516,2749],{"mathvariant":2748},[2542,13518,3174],{},[2542,13520,2749],{"mathvariant":2748},[3168,13522,13523,13525],{},[2542,13524,3793],{},[2542,13526,3293],{},[2689,13528,2756],{"stretchy":2755},[2689,13530,12273],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2542,13532,53],{"mathvariant":2748},[2549,13534,13535],{"encoding":2551},"\\begin{equation*}\n    \\langle Y\\rangle = \\text{Im}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{equation*}",[507,13537,13539],{"className":13538,"ariaHidden":2557},[2556],[507,13540,13542,13546],{"className":13541},[2561],[507,13543],{"className":13544,"style":13545},[2565],"height:1.8em;vertical-align:-0.65em;",[507,13547,13549],{"className":13548},[2570],[507,13550,13552],{"className":13551},[9452],[507,13553,13555],{"className":13554},[9711],[507,13556,13558,13710],{"className":13557},[2583,3200],[507,13559,13561,13707],{"className":13560},[2587],[507,13562,13564],{"className":13563,"style":12503},[2591],[507,13565,13566,13570],{"style":12506},[507,13567],{"className":13568,"style":13569},[2599],"height:3.15em;",[507,13571,13573,13576,13579,13582,13585,13588,13591,13597,13603,13606,13646,13649,13652,13655,13695,13698,13704],{"className":13572},[2570],[507,13574,4425],{"className":13575},[2941],[507,13577,7746],{"className":13578,"style":2715},[2570,2611],[507,13580,2756],{"className":13581},[2780],[507,13583],{"className":13584,"style":2919},[2714],[507,13586,573],{"className":13587},[2923],[507,13589],{"className":13590,"style":2919},[2714],[507,13592,13594],{"className":13593},[2570,7039],[507,13595,13504],{"className":13596},[2570],[507,13598,13600],{"className":13599},[2570],[507,13601,12248],{"className":13602},[2947,9920],[507,13604,4425],{"className":13605},[2941],[507,13607,13609,13612],{"className":13608},[2570],[507,13610,3793],{"className":13611,"style":2776},[2570,2611],[507,13613,13615],{"className":13614},[2579],[507,13616,13618,13638],{"className":13617},[2583,3200],[507,13619,13621,13635],{"className":13620},[2587],[507,13622,13624],{"className":13623,"style":6802},[2591],[507,13625,13626,13629],{"style":4572},[507,13627],{"className":13628,"style":2600},[2599],[507,13630,13632],{"className":13631},[2604,2605,2606,2607],[507,13633,2372],{"className":13634,"style":6823},[2570,2611,2607],[507,13636,3225],{"className":13637},[3224],[507,13639,13641],{"className":13640},[2587],[507,13642,13644],{"className":13643,"style":6833},[2591],[507,13645],{},[507,13647,2749],{"className":13648},[2570],[507,13650,3174],{"className":13651,"style":3220},[2570,2611],[507,13653,2749],{"className":13654},[2570],[507,13656,13658,13661],{"className":13657},[2570],[507,13659,3793],{"className":13660,"style":2776},[2570,2611],[507,13662,13664],{"className":13663},[2579],[507,13665,13667,13687],{"className":13666},[2583,3200],[507,13668,13670,13684],{"className":13669},[2587],[507,13671,13673],{"className":13672,"style":6802},[2591],[507,13674,13675,13678],{"style":4572},[507,13676],{"className":13677,"style":2600},[2599],[507,13679,13681],{"className":13680},[2604,2605,2606,2607],[507,13682,3293],{"className":13683},[2570,2611,2607],[507,13685,3225],{"className":13686},[3224],[507,13688,13690],{"className":13689},[2587],[507,13691,13693],{"className":13692,"style":3232},[2591],[507,13694],{},[507,13696,2756],{"className":13697},[2780],[507,13699,13701],{"className":13700},[2570],[507,13702,12273],{"className":13703},[2947,9920],[507,13705,53],{"className":13706},[2570],[507,13708,3225],{"className":13709},[3224],[507,13711,13713],{"className":13712},[2587],[507,13714,13716],{"className":13715,"style":12530},[2591],[507,13717],{},[498,13719,13721],{"className":500,"code":13720,"language":502,"meta":104,"style":104},"## Create the time-evo op circuit\nevol_gate = PauliEvolutionGate(\n    H_op, time=dt, synthesis=LieTrotter(reps=num_trotter_steps)\n)\n\n## Create the time-evo op dagger circuit\nevol_gate_d = PauliEvolutionGate(\n    H_op, time=dt, synthesis=LieTrotter(reps=num_trotter_steps)\n)\nevol_gate_d = evol_gate_d.inverse()\n\n# Put pieces together\nqc_reg = QuantumRegister(n_qubits)\nqc_temp = QuantumCircuit(qc_reg)\nqc_temp.compose(qc_state_prep, inplace=True)\nfor _ in range(num_trotter_steps):\n    qc_temp.append(evol_gate, qargs=qc_reg)\nfor _ in range(num_trotter_steps):\n    qc_temp.append(evol_gate_d, qargs=qc_reg)\nqc_temp.compose(qc_state_prep.inverse(), inplace=True)\n\n# Create controlled version of the circuit\ncontrolled_U = qc_temp.to_gate().control(1)\n\n# Create hadamard test circuit for real part\nqr = QuantumRegister(n_qubits + 1)\nqc_real = QuantumCircuit(qr)\nqc_real.h(0)\nqc_real.append(controlled_U, list(range(n_qubits + 1)))\nqc_real.h(0)\n\nprint(\n    \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n)\nqc_real.draw(\"mpl\", fold=-1, scale=0.5)\n",[504,13722,13723,13727,13737,13762,13766,13770,13775,13786,13810,13814,13828,13832,13837,13848,13860,13881,13894,13910,13922,13937,13959,13963,13968,13993,13997,14002,14018,14029,14042,14067,14079,14083,14089,14094,14098],{"__ignoreMap":104},[507,13724,13725],{"class":509,"line":510},[507,13726,9328],{"class":562},[507,13728,13729,13731,13733,13735],{"class":509,"line":105},[507,13730,9333],{"class":517},[507,13732,573],{"class":572},[507,13734,9338],{"class":576},[507,13736,1376],{"class":517},[507,13738,13739,13741,13743,13745,13748,13750,13752,13754,13756,13758,13760],{"class":509,"line":540},[507,13740,9345],{"class":517},[507,13742,9348],{"class":2155},[507,13744,573],{"class":572},[507,13746,13747],{"class":517},"dt, ",[507,13749,9356],{"class":2155},[507,13751,573],{"class":572},[507,13753,9361],{"class":576},[507,13755,580],{"class":517},[507,13757,9366],{"class":2155},[507,13759,573],{"class":572},[507,13761,9371],{"class":517},[507,13763,13764],{"class":509,"line":553},[507,13765,587],{"class":517},[507,13767,13768],{"class":509,"line":559},[507,13769,556],{"emptyLinePlaceholder":133},[507,13771,13772],{"class":509,"line":566},[507,13773,13774],{"class":562},"## Create the time-evo op dagger circuit\n",[507,13776,13777,13780,13782,13784],{"class":509,"line":590},[507,13778,13779],{"class":517},"evol_gate_d ",[507,13781,573],{"class":572},[507,13783,9338],{"class":576},[507,13785,1376],{"class":517},[507,13787,13788,13790,13792,13794,13796,13798,13800,13802,13804,13806,13808],{"class":509,"line":610},[507,13789,9345],{"class":517},[507,13791,9348],{"class":2155},[507,13793,573],{"class":572},[507,13795,13747],{"class":517},[507,13797,9356],{"class":2155},[507,13799,573],{"class":572},[507,13801,9361],{"class":576},[507,13803,580],{"class":517},[507,13805,9366],{"class":2155},[507,13807,573],{"class":572},[507,13809,9371],{"class":517},[507,13811,13812],{"class":509,"line":634},[507,13813,587],{"class":517},[507,13815,13816,13818,13820,13823,13826],{"class":509,"line":661},[507,13817,13779],{"class":517},[507,13819,573],{"class":572},[507,13821,13822],{"class":517}," evol_gate_d.",[507,13824,13825],{"class":576},"inverse",[507,13827,781],{"class":517},[507,13829,13830],{"class":509,"line":678},[507,13831,556],{"emptyLinePlaceholder":133},[507,13833,13834],{"class":509,"line":683},[507,13835,13836],{"class":562},"# Put pieces together\n",[507,13838,13839,13842,13844,13846],{"class":509,"line":697},[507,13840,13841],{"class":517},"qc_reg ",[507,13843,573],{"class":572},[507,13845,9389],{"class":576},[507,13847,8783],{"class":517},[507,13849,13850,13853,13855,13857],{"class":509,"line":710},[507,13851,13852],{"class":517},"qc_temp ",[507,13854,573],{"class":572},[507,13856,577],{"class":576},[507,13858,13859],{"class":517},"(qc_reg)\n",[507,13861,13862,13865,13868,13871,13874,13876,13879],{"class":509,"line":715},[507,13863,13864],{"class":517},"qc_temp.",[507,13866,13867],{"class":576},"compose",[507,13869,13870],{"class":517},"(qc_state_prep, ",[507,13872,13873],{"class":2155},"inplace",[507,13875,573],{"class":572},[507,13877,13878],{"class":583},"True",[507,13880,587],{"class":517},[507,13882,13883,13885,13887,13889,13891],{"class":509,"line":721},[507,13884,1630],{"class":513},[507,13886,8216],{"class":517},[507,13888,1636],{"class":513},[507,13890,8221],{"class":572},[507,13892,13893],{"class":517},"(num_trotter_steps):\n",[507,13895,13896,13899,13901,13903,13905,13907],{"class":509,"line":736},[507,13897,13898],{"class":517},"    qc_temp.",[507,13900,1939],{"class":576},[507,13902,9413],{"class":517},[507,13904,9416],{"class":2155},[507,13906,573],{"class":572},[507,13908,13909],{"class":517},"qc_reg)\n",[507,13911,13912,13914,13916,13918,13920],{"class":509,"line":748},[507,13913,1630],{"class":513},[507,13915,8216],{"class":517},[507,13917,1636],{"class":513},[507,13919,8221],{"class":572},[507,13921,13893],{"class":517},[507,13923,13924,13926,13928,13931,13933,13935],{"class":509,"line":761},[507,13925,13898],{"class":517},[507,13927,1939],{"class":576},[507,13929,13930],{"class":517},"(evol_gate_d, ",[507,13932,9416],{"class":2155},[507,13934,573],{"class":572},[507,13936,13909],{"class":517},[507,13938,13939,13941,13943,13946,13948,13951,13953,13955,13957],{"class":509,"line":775},[507,13940,13864],{"class":517},[507,13942,13867],{"class":576},[507,13944,13945],{"class":517},"(qc_state_prep.",[507,13947,13825],{"class":576},[507,13949,13950],{"class":517},"(), ",[507,13952,13873],{"class":2155},[507,13954,573],{"class":572},[507,13956,13878],{"class":583},[507,13958,587],{"class":517},[507,13960,13961],{"class":509,"line":784},[507,13962,556],{"emptyLinePlaceholder":133},[507,13964,13965],{"class":509,"line":796},[507,13966,13967],{"class":562},"# Create controlled version of the circuit\n",[507,13969,13970,13973,13975,13978,13981,13984,13987,13989,13991],{"class":509,"line":809},[507,13971,13972],{"class":517},"controlled_U ",[507,13974,573],{"class":572},[507,13976,13977],{"class":517}," qc_temp.",[507,13979,13980],{"class":576},"to_gate",[507,13982,13983],{"class":517},"().",[507,13985,13986],{"class":576},"control",[507,13988,580],{"class":517},[507,13990,625],{"class":583},[507,13992,587],{"class":517},[507,13994,13995],{"class":509,"line":1352},[507,13996,556],{"emptyLinePlaceholder":133},[507,13998,13999],{"class":509,"line":1357},[507,14000,14001],{"class":562},"# Create hadamard test circuit for real part\n",[507,14003,14004,14006,14008,14010,14012,14014,14016],{"class":509,"line":1362},[507,14005,9384],{"class":517},[507,14007,573],{"class":572},[507,14009,9389],{"class":576},[507,14011,2104],{"class":517},[507,14013,2107],{"class":572},[507,14015,1426],{"class":583},[507,14017,587],{"class":517},[507,14019,14020,14023,14025,14027],{"class":509,"line":1367},[507,14021,14022],{"class":517},"qc_real ",[507,14024,573],{"class":572},[507,14026,577],{"class":576},[507,14028,9403],{"class":517},[507,14030,14031,14034,14036,14038,14040],{"class":509,"line":1379},[507,14032,14033],{"class":517},"qc_real.",[507,14035,596],{"class":576},[507,14037,580],{"class":517},[507,14039,601],{"class":583},[507,14041,587],{"class":517},[507,14043,14044,14046,14048,14051,14054,14056,14058,14060,14062,14064],{"class":509,"line":1389},[507,14045,14033],{"class":517},[507,14047,1939],{"class":576},[507,14049,14050],{"class":517},"(controlled_U, ",[507,14052,14053],{"class":572},"list",[507,14055,580],{"class":517},[507,14057,2204],{"class":572},[507,14059,2104],{"class":517},[507,14061,2107],{"class":572},[507,14063,1426],{"class":583},[507,14065,14066],{"class":517},")))\n",[507,14068,14069,14071,14073,14075,14077],{"class":509,"line":1397},[507,14070,14033],{"class":517},[507,14072,596],{"class":576},[507,14074,580],{"class":517},[507,14076,601],{"class":583},[507,14078,587],{"class":517},[507,14080,14081],{"class":509,"line":1412},[507,14082,556],{"emptyLinePlaceholder":133},[507,14084,14085,14087],{"class":509,"line":1431},[507,14086,8525],{"class":572},[507,14088,1376],{"class":517},[507,14090,14091],{"class":509,"line":1449},[507,14092,14093],{"class":730},"    \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n",[507,14095,14096],{"class":509,"line":1465},[507,14097,587],{"class":517},[507,14099,14100,14102,14104,14106,14108,14110,14113,14116,14118,14120,14122,14124,14126],{"class":509,"line":1471},[507,14101,14033],{"class":517},[507,14103,9164],{"class":576},[507,14105,580],{"class":517},[507,14107,9169],{"class":730},[507,14109,622],{"class":517},[507,14111,14112],{"class":2155},"fold",[507,14114,14115],{"class":572},"=-",[507,14117,625],{"class":583},[507,14119,622],{"class":517},[507,14121,9174],{"class":2155},[507,14123,573],{"class":572},[507,14125,9179],{"class":583},[507,14127,587],{"class":517},[498,14129,14132],{"className":14130,"code":14131,"language":7039,"meta":104},[8531],"Circuit for calculating the real part of the overlap in S via Hadamard test\n",[504,14133,14131],{"__ignoreMap":104},[831,14135],{"alt":9184,"src":14136},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-02.avif",[18,14138,14139],{},"The Hadamard test circuit can be a deep circuit once we decompose to native gates (which will increase even more if we account for the topology of the device)",[498,14141,14143],{"className":500,"code":14142,"language":502,"meta":104,"style":104},"print(\n    \"Number of layers of 2Q operations\",\n    qc_real.decompose(reps=2).depth(lambda x: x[0].num_qubits == 2),\n)\n",[504,14144,14145,14151,14158,14203],{"__ignoreMap":104},[507,14146,14147,14149],{"class":509,"line":510},[507,14148,8525],{"class":572},[507,14150,1376],{"class":517},[507,14152,14153,14156],{"class":509,"line":105},[507,14154,14155],{"class":730},"    \"Number of layers of 2Q operations\"",[507,14157,1409],{"class":517},[507,14159,14160,14163,14166,14168,14170,14172,14174,14177,14180,14182,14185,14188,14191,14193,14196,14198,14200],{"class":509,"line":540},[507,14161,14162],{"class":517},"    qc_real.",[507,14164,14165],{"class":576},"decompose",[507,14167,580],{"class":517},[507,14169,9366],{"class":2155},[507,14171,573],{"class":572},[507,14173,584],{"class":583},[507,14175,14176],{"class":517},").",[507,14178,14179],{"class":576},"depth",[507,14181,580],{"class":517},[507,14183,14184],{"class":513},"lambda",[507,14186,14187],{"class":1382}," x",[507,14189,14190],{"class":517},": x[",[507,14192,601],{"class":583},[507,14194,14195],{"class":517},"].num_qubits ",[507,14197,1723],{"class":572},[507,14199,2316],{"class":583},[507,14201,14202],{"class":517},"),\n",[507,14204,14205],{"class":509,"line":553},[507,14206,587],{"class":517},[498,14208,14211],{"className":14209,"code":14210,"language":7039,"meta":104},[8531],"Number of layers of 2Q operations 112753\n",[504,14212,14210],{"__ignoreMap":104},[13,14214,14216],{"id":14215},"step-2-optimize-problem-for-quantum-hardware-execution","Step 2: Optimize problem for quantum hardware execution",[2513,14218,14220],{"id":14219},"efficient-hadamard-test","Efficient Hadamard test",[18,14222,14223],{},"We can optimize the deep circuits for the Hadamard test that we have obtained by introducing some approximations and relying on some assumption about the model Hamiltonian. For example, consider the following circuit for the Hadamard test:",[18,14225,14226],{},[6708,14227],{"alt":14228,"src":14229},"fig3.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-08.avif",[18,14231,14232,14233,14304,14305,14371,14372,14400,14401,14466,14467,14509,14510,14789,14790,14831,14832,14873],{},"Assume we can classically calculate ",[507,14234,14236,14254],{"className":14235},[2523],[507,14237,14239],{"className":14238},[2527],[2529,14240,14241],{"xmlns":2531},[2533,14242,14243,14251],{},[2536,14244,14245],{},[3168,14246,14247,14249],{},[2542,14248,6182],{},[2693,14250,601],{},[2549,14252,14253],{"encoding":2551},"E_0",[507,14255,14257],{"className":14256,"ariaHidden":2557},[2556],[507,14258,14260,14263],{"className":14259},[2561],[507,14261],{"className":14262,"style":3187},[2565],[507,14264,14266,14269],{"className":14265},[2570],[507,14267,6182],{"className":14268,"style":5892},[2570,2611],[507,14270,14272],{"className":14271},[2579],[507,14273,14275,14296],{"className":14274},[2583,3200],[507,14276,14278,14293],{"className":14277},[2587],[507,14279,14282],{"className":14280,"style":14281},[2591],"height:0.3011em;",[507,14283,14284,14287],{"style":6014},[507,14285],{"className":14286,"style":2600},[2599],[507,14288,14290],{"className":14289},[2604,2605,2606,2607],[507,14291,601],{"className":14292},[2570,2607],[507,14294,3225],{"className":14295},[3224],[507,14297,14299],{"className":14298},[2587],[507,14300,14302],{"className":14301,"style":3232},[2591],[507,14303],{},", the eigenvalue of ",[507,14306,14308,14330],{"className":14307},[2523],[507,14309,14311],{"className":14310},[2527],[2529,14312,14313],{"xmlns":2531},[2533,14314,14315,14327],{},[2536,14316,14317,14319,14321],{},[2542,14318,2749],{"mathvariant":2748},[2693,14320,601],{},[2539,14322,14323,14325],{},[2689,14324,2756],{"stretchy":2755},[2542,14326,7681],{},[2549,14328,14329],{"encoding":2551},"|0\\rangle^N",[507,14331,14333],{"className":14332,"ariaHidden":2557},[2556],[507,14334,14336,14339,14342],{"className":14335},[2561],[507,14337],{"className":14338,"style":5250},[2565],[507,14340,9735],{"className":14341},[2570],[507,14343,14345,14348],{"className":14344},[2780],[507,14346,2756],{"className":14347},[2780],[507,14349,14351],{"className":14350},[2579],[507,14352,14354],{"className":14353},[2583],[507,14355,14357],{"className":14356},[2587],[507,14358,14360],{"className":14359,"style":3330},[2591],[507,14361,14362,14365],{"style":2595},[507,14363],{"className":14364,"style":2600},[2599],[507,14366,14368],{"className":14367},[2604,2605,2606,2607],[507,14369,7681],{"className":14370,"style":3314},[2570,2611,2607]," under the Hamiltonian ",[507,14373,14375,14388],{"className":14374},[2523],[507,14376,14378],{"className":14377},[2527],[2529,14379,14380],{"xmlns":2531},[2533,14381,14382,14386],{},[2536,14383,14384],{},[2542,14385,3138],{},[2549,14387,3138],{"encoding":2551},[507,14389,14391],{"className":14390,"ariaHidden":2557},[2556],[507,14392,14394,14397],{"className":14393},[2561],[507,14395],{"className":14396,"style":2566},[2565],[507,14398,3138],{"className":14399,"style":3153},[2570,2611],".\nThis is satisfied when the Hamiltonian preserves the U(1) symmetry. Although this may seem like a strong assumption, there are many cases where it is safe to assume that there is a vacuum state (in this case it maps to the ",[507,14402,14404,14425],{"className":14403},[2523],[507,14405,14407],{"className":14406},[2527],[2529,14408,14409],{"xmlns":2531},[2533,14410,14411,14423],{},[2536,14412,14413,14415,14417],{},[2542,14414,2749],{"mathvariant":2748},[2693,14416,601],{},[2539,14418,14419,14421],{},[2689,14420,2756],{"stretchy":2755},[2542,14422,7681],{},[2549,14424,14329],{"encoding":2551},[507,14426,14428],{"className":14427,"ariaHidden":2557},[2556],[507,14429,14431,14434,14437],{"className":14430},[2561],[507,14432],{"className":14433,"style":5250},[2565],[507,14435,9735],{"className":14436},[2570],[507,14438,14440,14443],{"className":14439},[2780],[507,14441,2756],{"className":14442},[2780],[507,14444,14446],{"className":14445},[2579],[507,14447,14449],{"className":14448},[2583],[507,14450,14452],{"className":14451},[2587],[507,14453,14455],{"className":14454,"style":3330},[2591],[507,14456,14457,14460],{"style":2595},[507,14458],{"className":14459,"style":2600},[2599],[507,14461,14463],{"className":14462},[2604,2605,2606,2607],[507,14464,7681],{"className":14465,"style":3314},[2570,2611,2607]," state) which is unaffected by the action of the Hamiltonian. This is true for example for chemistry Hamiltonians that describe stable molecule (where the number of electrons is conserved).\nGiven that the gate ",[507,14468,14470,14488],{"className":14469},[2523],[507,14471,14473],{"className":14472},[2527],[2529,14474,14475],{"xmlns":2531},[2533,14476,14477,14485],{},[2536,14478,14479,14481,14483],{},[6167,14480,7013],{},[6167,14482,7016],{},[2542,14484,3793],{},[2549,14486,14487],{"encoding":2551},"\\text{Prep} \\; \\psi",[507,14489,14491],{"className":14490,"ariaHidden":2557},[2556],[507,14492,14494,14497,14503,14506],{"className":14493},[2561],[507,14495],{"className":14496,"style":7035},[2565],[507,14498,14500],{"className":14499},[2570,7039],[507,14501,7013],{"className":14502},[2570],[507,14504],{"className":14505,"style":2919},[2714],[507,14507,3793],{"className":14508,"style":2776},[2570,2611],", prepares the desired reference state ",[507,14511,14513,14591],{"className":14512},[2523],[507,14514,14516],{"className":14515},[2527],[2529,14517,14518],{"xmlns":2531},[2533,14519,14520,14588],{},[2536,14521,14522,14538,14540,14542,14544,14546,14554,14556,14574,14580],{},[14523,14524,14525,14527,14536],"mpadded",{},[2542,14526,2749],{"mathvariant":2748},[2536,14528,14529,14531,14534],{},[2542,14530,18],{},[2542,14532,14533],{},"s",[2542,14535,3293],{},[2689,14537,2756],{"stretchy":2755},[2689,14539,573],{},[6167,14541,7013],{},[6167,14543,7016],{},[2542,14545,3793],{},[14523,14547,14548,14550,14552],{},[2542,14549,2749],{"mathvariant":2748},[2693,14551,601],{},[2689,14553,2756],{"stretchy":2755},[2689,14555,573],{},[2539,14557,14558,14560],{},[2542,14559,3286],{},[2536,14561,14562,14564,14566,14568,14570,14572],{},[2689,14563,2691],{},[2542,14565,3293],{},[2542,14567,3138],{},[2693,14569,601],{},[2542,14571,4959],{},[2542,14573,3298],{},[3168,14575,14576,14578],{},[2542,14577,3279],{},[2542,14579,3793],{},[14523,14581,14582,14584,14586],{},[2542,14583,2749],{"mathvariant":2748},[2693,14585,601],{},[2689,14587,2756],{"stretchy":2755},[2549,14589,14590],{"encoding":2551},"\\ket{psi} = \\text{Prep} \\; \\psi \\ket{0} = e^{-i H 0 dt} U_{\\psi} \\ket{0}",[507,14592,14594,14630,14675],{"className":14593,"ariaHidden":2557},[2556],[507,14595,14597,14600,14621,14624,14627],{"className":14596},[2561],[507,14598],{"className":14599,"style":2769},[2565],[507,14601,14603,14606,14618],{"className":14602},[2937],[507,14604,2749],{"className":14605},[2570],[507,14607,14609,14612,14615],{"className":14608},[2570],[507,14610,18],{"className":14611},[2570,2611],[507,14613,14533],{"className":14614},[2570,2611],[507,14616,3293],{"className":14617},[2570,2611],[507,14619,2756],{"className":14620},[2780],[507,14622],{"className":14623,"style":2919},[2714],[507,14625,573],{"className":14626},[2923],[507,14628],{"className":14629,"style":2919},[2714],[507,14631,14633,14636,14642,14645,14648,14651,14666,14669,14672],{"className":14632},[2561],[507,14634],{"className":14635,"style":2769},[2565],[507,14637,14639],{"className":14638},[2570,7039],[507,14640,7013],{"className":14641},[2570],[507,14643],{"className":14644,"style":2919},[2714],[507,14646,3793],{"className":14647,"style":2776},[2570,2611],[507,14649],{"className":14650,"style":2965},[2714],[507,14652,14654,14657,14663],{"className":14653},[2937],[507,14655,2749],{"className":14656},[2570],[507,14658,14660],{"className":14659},[2570],[507,14661,601],{"className":14662},[2570],[507,14664,2756],{"className":14665},[2780],[507,14667],{"className":14668,"style":2919},[2714],[507,14670,573],{"className":14671},[2923],[507,14673],{"className":14674,"style":2919},[2714],[507,14676,14678,14681,14728,14771,14774],{"className":14677},[2561],[507,14679],{"className":14680,"style":11356},[2565],[507,14682,14684,14687],{"className":14683},[2570],[507,14685,3286],{"className":14686},[2570,2611],[507,14688,14690],{"className":14689},[2579],[507,14691,14693],{"className":14692},[2583],[507,14694,14696],{"className":14695},[2587],[507,14697,14699],{"className":14698,"style":3662},[2591],[507,14700,14701,14704],{"style":2595},[507,14702],{"className":14703,"style":2600},[2599],[507,14705,14707],{"className":14706},[2604,2605,2606,2607],[507,14708,14710,14713,14716,14719,14722,14725],{"className":14709},[2570,2607],[507,14711,2691],{"className":14712},[2570,2607],[507,14714,3293],{"className":14715},[2570,2611,2607],[507,14717,3138],{"className":14718,"style":3153},[2570,2611,2607],[507,14720,601],{"className":14721},[2570,2607],[507,14723,4959],{"className":14724},[2570,2611,2607],[507,14726,3298],{"className":14727},[2570,2611,2607],[507,14729,14731,14734],{"className":14730},[2570],[507,14732,3279],{"className":14733,"style":3314},[2570,2611],[507,14735,14737],{"className":14736},[2579],[507,14738,14740,14763],{"className":14739},[2583,3200],[507,14741,14743,14760],{"className":14742},[2587],[507,14744,14746],{"className":14745,"style":11243},[2591],[507,14747,14748,14751],{"style":11424},[507,14749],{"className":14750,"style":2600},[2599],[507,14752,14754],{"className":14753},[2604,2605,2606,2607],[507,14755,14757],{"className":14756},[2570,2607],[507,14758,3793],{"className":14759,"style":2776},[2570,2611,2607],[507,14761,3225],{"className":14762},[3224],[507,14764,14766],{"className":14765},[2587],[507,14767,14769],{"className":14768,"style":6833},[2591],[507,14770],{},[507,14772],{"className":14773,"style":2965},[2714],[507,14775,14777,14780,14786],{"className":14776},[2937],[507,14778,2749],{"className":14779},[2570],[507,14781,14783],{"className":14782},[2570],[507,14784,601],{"className":14785},[2570],[507,14787,2756],{"className":14788},[2780],", for example, to prepare the HF state for chemistry ",[507,14791,14793,14810],{"className":14792},[2523],[507,14794,14796],{"className":14795},[2527],[2529,14797,14798],{"xmlns":2531},[2533,14799,14800,14808],{},[2536,14801,14802,14804,14806],{},[6167,14803,7013],{},[6167,14805,7016],{},[2542,14807,3793],{},[2549,14809,14487],{"encoding":2551},[507,14811,14813],{"className":14812,"ariaHidden":2557},[2556],[507,14814,14816,14819,14825,14828],{"className":14815},[2561],[507,14817],{"className":14818,"style":7035},[2565],[507,14820,14822],{"className":14821},[2570,7039],[507,14823,7013],{"className":14824},[2570],[507,14826],{"className":14827,"style":2919},[2714],[507,14829,3793],{"className":14830,"style":2776},[2570,2611]," would be a product of single-qubit NOTs, so controlled-",[507,14833,14835,14852],{"className":14834},[2523],[507,14836,14838],{"className":14837},[2527],[2529,14839,14840],{"xmlns":2531},[2533,14841,14842,14850],{},[2536,14843,14844,14846,14848],{},[6167,14845,7013],{},[6167,14847,7016],{},[2542,14849,3793],{},[2549,14851,14487],{"encoding":2551},[507,14853,14855],{"className":14854,"ariaHidden":2557},[2556],[507,14856,14858,14861,14867,14870],{"className":14857},[2561],[507,14859],{"className":14860,"style":7035},[2565],[507,14862,14864],{"className":14863},[2570,7039],[507,14865,7013],{"className":14866},[2570],[507,14868],{"className":14869,"style":2919},[2714],[507,14871,3793],{"className":14872,"style":2776},[2570,2611]," is just a product of CNOTs.\nThen the circuit above implements the following state prior to measurement:",[507,14875,14877],{"className":14876},[2784],[507,14878,14880,15500],{"className":14879},[2523],[507,14881,14883],{"className":14882},[2527],[2529,14884,14885],{"xmlns":2531,"display":2793},[2533,14886,14887,15497],{},[9452,14888,14889],{"rowspacing":9454,"columnspacing":9455},[9457,14890,14891,14895,15490,15493],{},[9460,14892],{"className":14893},[14894],"mtr-glue",[9460,14896,14897],{},[9463,14898,14899],{"scriptlevel":601,"displaystyle":2557},[9452,14900,14901,15005,15072,15159,15242,15360],{"rowspacing":12114,"columnalign":12115,"columnspacing":12116},[9457,14902,14903,14941],{},[9460,14904,14905],{},[9463,14906,14907],{"scriptlevel":601,"displaystyle":2557},[2536,14908,14909,14917,14929],{},[14523,14910,14911,14913,14915],{},[2542,14912,2749],{"mathvariant":2748},[2693,14914,601],{},[2689,14916,2756],{"stretchy":2755},[2539,14918,14919,14927],{},[14523,14920,14921,14923,14925],{},[2542,14922,2749],{"mathvariant":2748},[2693,14924,601],{},[2689,14926,2756],{"stretchy":2755},[2542,14928,7681],{},[4271,14930,14931,14935],{},[2689,14932,14934],{"stretchy":2557,"minsize":14933},"3.0em","→",[14523,14936,14939],{"width":14937,"lspace":14938},"+0.6em","0.3em",[2542,14940,3138],{},[9460,14942,14943],{},[9463,14944,14945],{"scriptlevel":601,"displaystyle":2557},[2536,14946,14947,14949,14957],{},[2536,14948],{},[9491,14950,14951,14953],{},[2693,14952,625],{},[9496,14954,14955],{},[2693,14956,584],{},[2536,14958,14959,14961,14969,14981,14983,14991,15003],{},[2689,14960,580],{"fence":2557},[14523,14962,14963,14965,14967],{},[2542,14964,2749],{"mathvariant":2748},[2693,14966,601],{},[2689,14968,2756],{"stretchy":2755},[2539,14970,14971,14979],{},[14523,14972,14973,14975,14977],{},[2542,14974,2749],{"mathvariant":2748},[2693,14976,601],{},[2689,14978,2756],{"stretchy":2755},[2542,14980,7681],{},[2689,14982,2107],{},[14523,14984,14985,14987,14989],{},[2542,14986,2749],{"mathvariant":2748},[2693,14988,625],{},[2689,14990,2756],{"stretchy":2755},[2539,14992,14993,15001],{},[14523,14994,14995,14997,14999],{},[2542,14996,2749],{"mathvariant":2748},[2693,14998,601],{},[2689,15000,2756],{"stretchy":2755},[2542,15002,7681],{},[2689,15004,3649],{"fence":2557},[9457,15006,15007,15020],{},[9460,15008,15009],{},[9463,15010,15011],{"scriptlevel":601,"displaystyle":2557},[4271,15012,15013,15015],{},[2689,15014,14934],{"stretchy":2557,"minsize":14933},[14523,15016,15017],{"width":14937,"lspace":14938},[6167,15018,15019],{},"1-ctrl-init",[9460,15021,15022],{},[9463,15023,15024],{"scriptlevel":601,"displaystyle":2557},[2536,15025,15026,15028,15036],{},[2536,15027],{},[9491,15029,15030,15032],{},[2693,15031,625],{},[9496,15033,15034],{},[2693,15035,584],{},[2536,15037,15038,15040,15042,15044,15046,15048,15050,15056,15058,15060,15062,15064,15066,15068,15070],{},[2689,15039,580],{"fence":2557},[2542,15041,2749],{"mathvariant":2748},[2693,15043,601],{},[2689,15045,2756],{"stretchy":2755},[2542,15047,2749],{"mathvariant":2748},[2693,15049,601],{},[2539,15051,15052,15054],{},[2689,15053,2756],{"stretchy":2755},[2542,15055,7681],{},[2689,15057,2107],{},[2542,15059,2749],{"mathvariant":2748},[2693,15061,625],{},[2689,15063,2756],{"stretchy":2755},[2542,15065,2749],{"mathvariant":2748},[2542,15067,3793],{},[2689,15069,2756],{"stretchy":2755},[2689,15071,3649],{"fence":2557},[9457,15073,15074,15086],{},[9460,15075,15076],{},[9463,15077,15078],{"scriptlevel":601,"displaystyle":2557},[4271,15079,15080,15082],{},[2689,15081,14934],{"stretchy":2557,"minsize":14933},[14523,15083,15084],{"width":14937,"lspace":14938},[2542,15085,3279],{},[9460,15087,15088],{},[9463,15089,15090],{"scriptlevel":601,"displaystyle":2557},[2536,15091,15092,15094,15102],{},[2536,15093],{},[9491,15095,15096,15098],{},[2693,15097,625],{},[9496,15099,15100],{},[2693,15101,584],{},[2536,15103,15104,15106,15117,15125,15137,15139,15147,15149,15157],{},[2689,15105,580],{"fence":2557},[2539,15107,15108,15110],{},[2542,15109,3286],{},[2536,15111,15112,15114],{},[2542,15113,3293],{},[2542,15115,15116],{},"ϕ",[14523,15118,15119,15121,15123],{},[2542,15120,2749],{"mathvariant":2748},[2693,15122,601],{},[2689,15124,2756],{"stretchy":2755},[2539,15126,15127,15135],{},[14523,15128,15129,15131,15133],{},[2542,15130,2749],{"mathvariant":2748},[2693,15132,601],{},[2689,15134,2756],{"stretchy":2755},[2542,15136,7681],{},[2689,15138,2107],{},[14523,15140,15141,15143,15145],{},[2542,15142,2749],{"mathvariant":2748},[2693,15144,625],{},[2689,15146,2756],{"stretchy":2755},[2542,15148,3279],{},[14523,15150,15151,15153,15155],{},[2542,15152,2749],{"mathvariant":2748},[2542,15154,3793],{},[2689,15156,2756],{"stretchy":2755},[2689,15158,3649],{"fence":2557},[9457,15160,15161,15174],{},[9460,15162,15163],{},[9463,15164,15165],{"scriptlevel":601,"displaystyle":2557},[4271,15166,15167,15169],{},[2689,15168,14934],{"stretchy":2557,"minsize":14933},[14523,15170,15171],{"width":14937,"lspace":14938},[6167,15172,15173],{},"0-ctrl-init",[9460,15175,15176],{},[9463,15177,15178],{"scriptlevel":601,"displaystyle":2557},[2536,15179,15180,15182,15190],{},[2536,15181],{},[9491,15183,15184,15186],{},[2693,15185,625],{},[9496,15187,15188],{},[2693,15189,584],{},[2536,15191,15192,15194,15204,15212,15220,15222,15230,15232,15240],{},[2689,15193,580],{"fence":2557},[2539,15195,15196,15198],{},[2542,15197,3286],{},[2536,15199,15200,15202],{},[2542,15201,3293],{},[2542,15203,15116],{},[14523,15205,15206,15208,15210],{},[2542,15207,2749],{"mathvariant":2748},[2693,15209,601],{},[2689,15211,2756],{"stretchy":2755},[14523,15213,15214,15216,15218],{},[2542,15215,2749],{"mathvariant":2748},[2542,15217,3793],{},[2689,15219,2756],{"stretchy":2755},[2689,15221,2107],{},[14523,15223,15224,15226,15228],{},[2542,15225,2749],{"mathvariant":2748},[2693,15227,625],{},[2689,15229,2756],{"stretchy":2755},[2542,15231,3279],{},[14523,15233,15234,15236,15238],{},[2542,15235,2749],{"mathvariant":2748},[2542,15237,3793],{},[2689,15239,2756],{"stretchy":2755},[2689,15241,3649],{"fence":2557},[9457,15243,15244,15250],{},[9460,15245,15246],{},[9463,15247,15248],{"scriptlevel":601,"displaystyle":2557},[2689,15249,573],{"lspace":12116,"rspace":12116},[9460,15251,15252],{},[9463,15253,15254],{"scriptlevel":601,"displaystyle":2557},[2536,15255,15256,15258,15264],{},[2536,15257],{},[9491,15259,15260,15262],{},[2693,15261,625],{},[2693,15263,584],{},[2536,15265,15266,15268,15276,15312,15314,15322,15358],{},[2689,15267,580],{"fence":2557},[14523,15269,15270,15272,15274],{},[2542,15271,2749],{"mathvariant":2748},[2689,15273,2107],{"lspace":12116,"rspace":12116},[2689,15275,2756],{"stretchy":2755},[2536,15277,15278,15280,15290,15298,15300,15302,15310],{},[2689,15279,580],{"fence":2557},[2539,15281,15282,15284],{},[2542,15283,3286],{},[2536,15285,15286,15288],{},[2542,15287,3293],{},[2542,15289,15116],{},[14523,15291,15292,15294,15296],{},[2542,15293,2749],{"mathvariant":2748},[2542,15295,3793],{},[2689,15297,2756],{"stretchy":2755},[2689,15299,2107],{},[2542,15301,3279],{},[14523,15303,15304,15306,15308],{},[2542,15305,2749],{"mathvariant":2748},[2542,15307,3793],{},[2689,15309,2756],{"stretchy":2755},[2689,15311,3649],{"fence":2557},[2689,15313,2107],{},[14523,15315,15316,15318,15320],{},[2542,15317,2749],{"mathvariant":2748},[2689,15319,2691],{"lspace":12116,"rspace":12116},[2689,15321,2756],{"stretchy":2755},[2536,15323,15324,15326,15336,15344,15346,15348,15356],{},[2689,15325,580],{"fence":2557},[2539,15327,15328,15330],{},[2542,15329,3286],{},[2536,15331,15332,15334],{},[2542,15333,3293],{},[2542,15335,15116],{},[14523,15337,15338,15340,15342],{},[2542,15339,2749],{"mathvariant":2748},[2542,15341,3793],{},[2689,15343,2756],{"stretchy":2755},[2689,15345,2691],{},[2542,15347,3279],{},[14523,15349,15350,15352,15354],{},[2542,15351,2749],{"mathvariant":2748},[2542,15353,3793],{},[2689,15355,2756],{"stretchy":2755},[2689,15357,3649],{"fence":2557},[2689,15359,3649],{"fence":2557},[9457,15361,15362,15368],{},[9460,15363,15364],{},[9463,15365,15366],{"scriptlevel":601,"displaystyle":2557},[2689,15367,573],{"lspace":12116,"rspace":12116},[9460,15369,15370],{},[9463,15371,15372],{"scriptlevel":601,"displaystyle":2557},[2536,15373,15374,15376,15382],{},[2536,15375],{},[9491,15377,15378,15380],{},[2693,15379,625],{},[2693,15381,584],{},[2536,15383,15384,15386,15398,15436,15438,15450,15488],{},[2689,15385,580],{"fence":2557},[14523,15387,15388,15390,15396],{},[2542,15389,2749],{"mathvariant":2748},[2536,15391,15392,15394],{},[2689,15393,2107],{},[2542,15395,3293],{},[2689,15397,2756],{"stretchy":2755},[2536,15399,15400,15402,15412,15420,15422,15424,15426,15434],{},[2689,15401,580],{"fence":2557},[2539,15403,15404,15406],{},[2542,15405,3286],{},[2536,15407,15408,15410],{},[2542,15409,3293],{},[2542,15411,15116],{},[14523,15413,15414,15416,15418],{},[2542,15415,2749],{"mathvariant":2748},[2542,15417,3793],{},[2689,15419,2756],{"stretchy":2755},[2689,15421,2691],{},[2542,15423,3293],{},[2542,15425,3279],{},[14523,15427,15428,15430,15432],{},[2542,15429,2749],{"mathvariant":2748},[2542,15431,3793],{},[2689,15433,2756],{"stretchy":2755},[2689,15435,3649],{"fence":2557},[2689,15437,2107],{},[14523,15439,15440,15442,15448],{},[2542,15441,2749],{"mathvariant":2748},[2536,15443,15444,15446],{},[2689,15445,2691],{},[2542,15447,3293],{},[2689,15449,2756],{"stretchy":2755},[2536,15451,15452,15454,15464,15472,15474,15476,15478,15486],{},[2689,15453,580],{"fence":2557},[2539,15455,15456,15458],{},[2542,15457,3286],{},[2536,15459,15460,15462],{},[2542,15461,3293],{},[2542,15463,15116],{},[14523,15465,15466,15468,15470],{},[2542,15467,2749],{"mathvariant":2748},[2542,15469,3793],{},[2689,15471,2756],{"stretchy":2755},[2689,15473,2107],{},[2542,15475,3293],{},[2542,15477,3279],{},[14523,15479,15480,15482,15484],{},[2542,15481,2749],{"mathvariant":2748},[2542,15483,3793],{},[2689,15485,2756],{"stretchy":2755},[2689,15487,3649],{"fence":2557},[2689,15489,3649],{"fence":2557},[9460,15491],{"className":15492},[14894],[9460,15494],{"className":15495},[15496],"mml-eqn-num",[2549,15498,15499],{"encoding":2551},"\\begin{equation}\n\\begin{split}\n    \\ket{0} \\ket{0}^N\\xrightarrow{H}&\\frac{1}{\\sqrt{2}}\n    \\left(\n    \\ket{0}\\ket{0}^N+ \\ket{1} \\ket{0}^N\n    \\right)\\\\\n    \\xrightarrow{\\text{1-ctrl-init}}&\\frac{1}{\\sqrt{2}}\\left(|0\\rangle|0\\rangle^N+|1\\rangle|\\psi\\rangle\\right)\\\\\n    \\xrightarrow{U}&\\frac{1}{\\sqrt{2}}\\left(e^{i\\phi}\\ket{0}\\ket{0}^N+\\ket{1} U\\ket{\\psi}\\right)\\\\\n    \\xrightarrow{\\text{0-ctrl-init}}&\\frac{1}{\\sqrt{2}}\n    \\left(\n    e^{i\\phi}\\ket{0} \\ket{\\psi}\n    +\\ket{1} U\\ket{\\psi}\n    \\right)\\\\\n    =&\\frac{1}{2}\n    \\left(\n    \\ket{+}\\left(e^{i\\phi}\\ket{\\psi}+U\\ket{\\psi}\\right)\n    +\\ket{-}\\left(e^{i\\phi}\\ket{\\psi}-U\\ket{\\psi}\\right)\n    \\right)\\\\\n    =&\\frac{1}{2}\n    \\left(\n    \\ket{+i}\\left(e^{i\\phi}\\ket{\\psi}-iU\\ket{\\psi}\\right)\n    +\\ket{-i}\\left(e^{i\\phi}\\ket{\\psi}+iU\\ket{\\psi}\\right)\n    \\right)\n\\end{split}\n\\end{equation}",[507,15501,15503,17608],{"className":15502,"ariaHidden":2557},[2556],[507,15504,15506,15510],{"className":15505},[2561],[507,15507],{"className":15508,"style":15509},[2565],"height:14.5206em;vertical-align:-7.0103em;",[507,15511,15513],{"className":15512},[9452],[507,15514,15516],{"className":15515},[9711],[507,15517,15519,17600],{"className":15518},[2583,3200],[507,15520,15522,17597],{"className":15521},[2587],[507,15523,15526],{"className":15524,"style":15525},[2591],"height:7.5103em;",[507,15527,15529,15533],{"style":15528},"top:-9.5103em;",[507,15530],{"className":15531,"style":15532},[2599],"height:9.5103em;",[507,15534,15536],{"className":15535},[2570],[507,15537,15539],{"className":15538},[2570],[507,15540,15542,15905],{"className":15541},[9452],[507,15543,15545],{"className":15544},[12327],[507,15546,15548,15896],{"className":15547},[2583,3200],[507,15549,15551,15893],{"className":15550},[2587],[507,15552,15554,15685,15748,15807,15869,15881],{"className":15553,"style":15525},[2591],[507,15555,15556,15559],{"style":15528},[507,15557],{"className":15558,"style":9728},[2599],[507,15560,15562,15577,15580,15621,15624],{"className":15561},[2570],[507,15563,15565,15568,15574],{"className":15564},[2937],[507,15566,2749],{"className":15567},[2570],[507,15569,15571],{"className":15570},[2570],[507,15572,601],{"className":15573},[2570],[507,15575,2756],{"className":15576},[2780],[507,15578],{"className":15579,"style":2965},[2714],[507,15581,15583,15598],{"className":15582},[2937],[507,15584,15586,15589,15595],{"className":15585},[2937],[507,15587,2749],{"className":15588},[2570],[507,15590,15592],{"className":15591},[2570],[507,15593,601],{"className":15594},[2570],[507,15596,2756],{"className":15597},[2780],[507,15599,15601],{"className":15600},[2579],[507,15602,15604],{"className":15603},[2583],[507,15605,15607],{"className":15606},[2587],[507,15608,15610],{"className":15609,"style":7822},[2591],[507,15611,15612,15615],{"style":7845},[507,15613],{"className":15614,"style":2600},[2599],[507,15616,15618],{"className":15617},[2604,2605,2606,2607],[507,15619,7681],{"className":15620,"style":3314},[2570,2611,2607],[507,15622],{"className":15623,"style":2919},[2714],[507,15625,15628],{"className":15626},[2923,15627],"x-arrow",[507,15629,15631,15676],{"className":15630},[2583,3200],[507,15632,15634,15673],{"className":15633},[2587],[507,15635,15638,15654],{"className":15636,"style":15637},[2591],"height:1.1003em;",[507,15639,15641,15644],{"style":15640},"top:-3.322em;",[507,15642],{"className":15643,"style":2600},[2599],[507,15645,15648],{"className":15646},[2604,2605,2606,2607,15647],"x-arrow-pad",[507,15649,15651],{"className":15650},[2570,2607],[507,15652,3138],{"className":15653,"style":3153},[2570,2611,2607],[507,15655,15658,15661],{"className":15656,"style":15657},[9833],"top:-2.689em;",[507,15659],{"className":15660,"style":2600},[2599],[507,15662,15665],{"className":15663,"style":15664},[9853],"height:0.522em;min-width:1.469em;",[6281,15666,15670],{"xmlns":6283,"width":9857,"height":15667,"viewBox":15668,"preserveAspectRatio":15669},"0.522em","0 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we have used the classical simulable phase shift ",[507,17645,17647,17697],{"className":17646},[2523],[507,17648,17650],{"className":17649},[2527],[2529,17651,17652],{"xmlns":2531},[2533,17653,17654,17694],{},[2536,17655,17656,17658,17670,17672,17682],{},[2542,17657,3279],{},[2539,17659,17660,17668],{},[14523,17661,17662,17664,17666],{},[2542,17663,2749],{"mathvariant":2748},[2693,17665,601],{},[2689,17667,2756],{"stretchy":2755},[2542,17669,7681],{},[2689,17671,573],{},[2539,17673,17674,17676],{},[2542,17675,3286],{},[2536,17677,17678,17680],{},[2542,17679,3293],{},[2542,17681,15116],{},[2539,17683,17684,17692],{},[14523,17685,17686,17688,17690],{},[2542,17687,2749],{"mathvariant":2748},[2693,17689,601],{},[2689,17691,2756],{"stretchy":2755},[2542,17693,7681],{},[2549,17695,17696],{"encoding":2551}," U\\ket{0}^N = e^{i\\phi}\\ket{0}^N",[507,17698,17700,17763],{"className":17699,"ariaHidden":2557},[2556],[507,17701,17703,17707,17710,17713,17754,17757,17760],{"className":17702},[2561],[507,17704],{"className":17705,"style":17706},[2565],"height:1.2312em;vertical-align:-0.25em;",[507,17708,3279],{"className":17709,"style":3314},[2570,2611],[507,17711],{"className":17712,"style":2965},[2714],[507,17714,17716,17731],{"className":17715},[2937],[507,17717,17719,17722,17728],{"className":17718},[2937],[507,17720,2749],{"className":17721},[2570],[507,17723,17725],{"className":17724},[2570],[507,17726,601],{"className":17727},[2570],[507,17729,2756],{"className":17730},[2780],[507,17732,17734],{"className":17733},[2579],[507,17735,17737],{"className":17736},[2583],[507,17738,17740],{"className":17739},[2587],[507,17741,17743],{"className":17742,"style":7822},[2591],[507,17744,17745,17748],{"style":7845},[507,17746],{"className":17747,"style":2600},[2599],[507,17749,17751],{"className":17750},[2604,2605,2606,2607],[507,17752,7681],{"className":17753,"style":3314},[2570,2611,2607],[507,17755],{"className":17756,"style":2919},[2714],[507,17758,573],{"className":17759},[2923],[507,17761],{"className":17762,"style":2919},[2714],[507,17764,17766,17769,17804,17807],{"className":17765},[2561],[507,17767],{"className":17768,"style":17706},[2565],[507,17770,17772,17775],{"className":17771},[2570],[507,17773,3286],{"className":17774},[2570,2611],[507,17776,17778],{"className":17777},[2579],[507,17779,17781],{"className":17780},[2583],[507,17782,17784],{"className":17783},[2587],[507,17785,17787],{"className":17786,"style":3662},[2591],[507,17788,17789,17792],{"style":2595},[507,17790],{"className":17791,"style":2600},[2599],[507,17793,17795],{"className":17794},[2604,2605,2606,2607],[507,17796,17798,17801],{"className":17797},[2570,2607],[507,17799,3293],{"className":17800},[2570,2611,2607],[507,17802,15116],{"className":17803},[2570,2611,2607],[507,17805],{"className":17806,"style":2965},[2714],[507,17808,17810,17825],{"className":17809},[2937],[507,17811,17813,17816,17822],{"className":17812},[2937],[507,17814,2749],{"className":17815},[2570],[507,17817,17819],{"className":17818},[2570],[507,17820,601],{"className":17821},[2570],[507,17823,2756],{"className":17824},[2780],[507,17826,17828],{"className":17827},[2579],[507,17829,17831],{"className":17830},[2583],[507,17832,17834],{"className":17833},[2587],[507,17835,17837],{"className":17836,"style":7822},[2591],[507,17838,17839,17842],{"style":7845},[507,17840],{"className":17841,"style":2600},[2599],[507,17843,17845],{"className":17844},[2604,2605,2606,2607],[507,17846,7681],{"className":17847,"style":3314},[2570,2611,2607]," in the third line. Therefore the expectation values are obtained as",[507,17850,17852],{"className":17851},[2784],[507,17853,17855,18165],{"className":17854},[2523],[507,17856,17858],{"className":17857},[2527],[2529,17859,17860],{"xmlns":2531,"display":2793},[2533,17861,17862,18162],{},[9452,17863,17864],{"rowspacing":9454,"columnspacing":9455},[9457,17865,17866,17869,18156,18159],{},[9460,17867],{"className":17868},[14894],[9460,17870,17871],{},[9463,17872,17873],{"scriptlevel":601,"displaystyle":2557},[9452,17874,17875,17993,18096],{"rowspacing":12114,"columnalign":12115,"columnspacing":12116},[9457,17876,17877,17894],{},[9460,17878,17879],{},[9463,17880,17881],{"scriptlevel":601,"displaystyle":2557},[2536,17882,17883,17885,17887,17890,17892],{},[2689,17884,4425],{"stretchy":2755},[2542,17886,7731],{},[2689,17888,17889],{},"⊗",[2542,17891,3174],{},[2689,17893,2756],{"stretchy":2755},[9460,17895,17896],{},[9463,17897,17898],{"scriptlevel":601,"displaystyle":2557},[2536,17899,17900,17902,17904,17910,17912,17955,17957],{},[2536,17901],{},[2689,17903,573],{},[9491,17905,17906,17908],{},[2693,17907,625],{},[2693,17909,12152],{},[2689,17911,580],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[2536,17913,17914,17916,17928,17936,17938,17946,17953],{},[2689,17915,580],{"fence":2557},[2539,17917,17918,17920],{},[2542,17919,3286],{},[2536,17921,17922,17924,17926],{},[2689,17923,2691],{},[2542,17925,3293],{},[2542,17927,15116],{},[14523,17929,17930,17932,17934],{},[2689,17931,4425],{"stretchy":2755},[2542,17933,3793],{},[2542,17935,2749],{"mathvariant":2748},[2689,17937,2107],{},[14523,17939,17940,17942,17944],{},[2689,17941,4425],{"stretchy":2755},[2542,17943,3793],{},[2542,17945,2749],{"mathvariant":2748},[2539,17947,17948,17950],{},[2542,17949,3279],{},[2689,17951,17952],{},"†",[2689,17954,3649],{"fence":2557},[2542,17956,3174],{},[2536,17958,17959,17961,17971,17979,17981,17983,17991],{},[2689,17960,580],{"fence":2557},[2539,17962,17963,17965],{},[2542,17964,3286],{},[2536,17966,17967,17969],{},[2542,17968,3293],{},[2542,17970,15116],{},[14523,17972,17973,17975,17977],{},[2542,17974,2749],{"mathvariant":2748},[2542,17976,3793],{},[2689,17978,2756],{"stretchy":2755},[2689,17980,2107],{},[2542,17982,3279],{},[14523,17984,17985,17987,17989],{},[2542,17986,2749],{"mathvariant":2748},[2542,17988,3793],{},[2689,17990,2756],{"stretchy":2755},[2689,17992,3649],{"fence":2557},[9457,17994,17995,18001],{},[9460,17996,17997],{},[9463,17998,17999],{"scriptlevel":601,"displaystyle":2557},[2536,18000],{},[9460,18002,18003],{},[9463,18004,18005],{"scriptlevel":601,"displaystyle":2557},[2536,18006,18007,18009,18012,18014,18056,18058,18094],{},[2536,18008],{},[2714,18010],{"width":18011},"2em",[2689,18013,2691],{},[2536,18015,18016,18018,18030,18038,18040,18048,18054],{},[2689,18017,580],{"fence":2557},[2539,18019,18020,18022],{},[2542,18021,3286],{},[2536,18023,18024,18026,18028],{},[2689,18025,2691],{},[2542,18027,3293],{},[2542,18029,15116],{},[14523,18031,18032,18034,18036],{},[2689,18033,4425],{"stretchy":2755},[2542,18035,3793],{},[2542,18037,2749],{"mathvariant":2748},[2689,18039,2691],{},[14523,18041,18042,18044,18046],{},[2689,18043,4425],{"stretchy":2755},[2542,18045,3793],{},[2542,18047,2749],{"mathvariant":2748},[2539,18049,18050,18052],{},[2542,18051,3279],{},[2689,18053,17952],{},[2689,18055,3649],{"fence":2557},[2542,18057,3174],{},[2536,18059,18060,18062,18072,18080,18082,18084,18092],{},[2689,18061,580],{"fence":2557},[2539,18063,18064,18066],{},[2542,18065,3286],{},[2536,18067,18068,18070],{},[2542,18069,3293],{},[2542,18071,15116],{},[14523,18073,18074,18076,18078],{},[2542,18075,2749],{"mathvariant":2748},[2542,18077,3793],{},[2689,18079,2756],{"stretchy":2755},[2689,18081,2691],{},[2542,18083,3279],{},[14523,18085,18086,18088,18090],{},[2542,18087,2749],{"mathvariant":2748},[2542,18089,3793],{},[2689,18091,2756],{"stretchy":2755},[2689,18093,3649],{"fence":2557},[2689,18095,3649],{"fence":2755,"stretchy":2557,"minsize":9502,"maxsize":9502},[9457,18097,18098,18104],{},[9460,18099,18100],{},[9463,18101,18102],{"scriptlevel":601,"displaystyle":2557},[2536,18103],{},[9460,18105,18106],{},[9463,18107,18108],{"scriptlevel":601,"displaystyle":2557},[2536,18109,18110,18112,18114,18116,18154],{},[2536,18111],{},[2689,18113,573],{},[6167,18115,12245],{},[2536,18117,18118,18120,18132,18140,18142,18144,18152],{},[2689,18119,12248],{"fence":2557},[2539,18121,18122,18124],{},[2542,18123,3286],{},[2536,18125,18126,18128,18130],{},[2689,18127,2691],{},[2542,18129,3293],{},[2542,18131,15116],{},[14523,18133,18134,18136,18138],{},[2689,18135,4425],{"stretchy":2755},[2542,18137,3793],{},[2542,18139,2749],{"mathvariant":2748},[2542,18141,3174],{},[2542,18143,3279],{},[14523,18145,18146,18148,18150],{},[2542,18147,2749],{"mathvariant":2748},[2542,18149,3793],{},[2689,18151,2756],{"stretchy":2755},[2689,18153,12273],{"fence":2557},[2689,18155,2819],{"separator":2557},[9460,18157],{"className":18158},[14894],[9460,18160],{"className":18161},[15496],[2549,18163,18164],{"encoding":2551},"\\begin{equation}\n\\begin{split}\n    \\langle X\\otimes P\\rangle&=\\frac{1}{4}\n    \\Big(\n    \\left(e^{-i\\phi}\\bra{\\psi}+\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}+U\\ket{\\psi}\\right)\n    \\\\\n    &\\qquad-\\left(e^{-i\\phi}\\bra{\\psi}-\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}-U\\ket{\\psi}\\right)\n    \\Big)\\\\\n    &=\\text{Re}\\left[e^{-i\\phi}\\bra{\\psi}PU\\ket{\\psi}\\right],\n\\end{split}\n\\end{equation}",[507,18166,18168,19037],{"className":18167,"ariaHidden":2557},[2556],[507,18169,18171,18175],{"className":18170},[2561],[507,18172],{"className":18173,"style":18174},[2565],"height:5.6666em;vertical-align:-2.5833em;",[507,18176,18178],{"className":18177},[9452],[507,18179,18181],{"className":18180},[9711],[507,18182,18184,19029],{"className":18183},[2583,3200],[507,18185,18187,19026],{"className":18186},[2587],[507,18188,18191],{"className":18189,"style":18190},[2591],"height:3.0833em;",[507,18192,18194,18198],{"style":18193},"top:-5.0833em;",[507,18195],{"className":18196,"style":18197},[2599],"height:5.0833em;",[507,18199,18201],{"className":18200},[2570],[507,18202,18204],{"className":18203},[2570],[507,18205,18207,18278],{"className":18206},[9452],[507,18208,18210],{"className":18209},[12327],[507,18211,18213,18269],{"className":18212},[2583,3200],[507,18214,18216,18266],{"className":18215},[2587],[507,18217,18219,18248,18257],{"className":18218,"style":18190},[2591],[507,18220,18221,18224],{"style":18193},[507,18222],{"className":18223,"style":9728},[2599],[507,18225,18227,18230,18233,18236,18239,18242,18245],{"className":18226},[2570],[507,18228,4425],{"className":18229},[2941],[507,18231,7731],{"className":18232,"style":7876},[2570,2611],[507,18234],{"className":18235,"style":2715},[2714],[507,18237,17889],{"className":18238},[2719],[507,18240],{"className":18241,"style":2715},[2714],[507,18243,3174],{"className":18244,"style":3220},[2570,2611],[507,18246,2756],{"className":18247},[2780],[507,18249,18251,18254],{"style":18250},"top:-2.9473em;",[507,18252],{"className":18253,"style":9728},[2599],[507,18255],{"className":18256},[2570],[507,18258,18260,18263],{"style":18259},"top:-1.0982em;",[507,18261],{"className":18262,"style":9728},[2599],[507,18264],{"className":18265},[2570],[507,18267,3225],{"className":18268},[3224],[507,18270,18272],{"className":18271},[2587],[507,18273,18276],{"className":18274,"style":18275},[2591],"height:2.5833em;",[507,18277],{},[507,18279,18281],{"className":18280},[12388],[507,18282,18284,19018],{"className":18283},[2583,3200],[507,18285,18287,19015],{"className":18286},[2587],[507,18288,18290,18615,18882],{"className":18289,"style":18190},[2591],[507,18291,18292,18295],{"style":18193},[507,18293],{"className":18294,"style":9728},[2599],[507,18296,18298,18301,18304,18307,18310,18372,18378,18381,18508,18511,18514,18517],{"className":18297},[2570],[507,18299],{"className":18300},[2570],[507,18302],{"className":18303,"style":2919},[2714],[507,18305,573],{"className":18306},[2923],[507,18308],{"className":18309,"style":2919},[2714],[507,18311,18313,18316,18369],{"className":18312},[2570],[507,18314],{"className":18315},[2941,9793],[507,18317,18319],{"className":18318},[9491],[507,18320,18322,18361],{"className":18321},[2583,3200],[507,18323,18325,18358],{"className":18324},[2587],[507,18326,18328,18339,18347],{"className":18327,"style":9806},[2591],[507,18329,18330,18333],{"style":11717},[507,18331],{"className":18332,"style":4310},[2599],[507,18334,18336],{"className":18335},[2570],[507,18337,12152],{"className":18338},[2570],[507,18340,18341,18344],{"style":9878},[507,18342],{"className":18343,"style":4310},[2599],[507,18345],{"className":18346,"style":9886},[9885],[507,18348,18349,18352],{"style":9889},[507,18350],{"className":18351,"style":4310},[2599],[507,18353,18355],{"className":18354},[2570],[507,18356,625],{"className":18357},[2570],[507,18359,3225],{"className":18360},[3224],[507,18362,18364],{"className":18363},[2587],[507,18365,18367],{"className":18366,"style":11755},[2591],[507,18368],{},[507,18370],{"className":18371},[2780,9793],[507,18373,18375],{"className":18374},[2570],[507,18376,580],{"className":18377},[2947,9920],[507,18379],{"className":18380,"style":2965},[2714],[507,18382,18384,18390,18428,18431,18446,18449,18452,18455,18470,18473,18502],{"className":18383},[2937],[507,18385,18387],{"className":18386,"style":2943},[2941,2942],[507,18388,580],{"className":18389},[2947,2948],[507,18391,18393,18396],{"className":18392},[2570],[507,18394,3286],{"className":18395},[2570,2611],[507,18397,18399],{"className":18398},[2579],[507,18400,18402],{"className":18401},[2583],[507,18403,18405],{"className":18404},[2587],[507,18406,18408],{"className":18407,"style":5040},[2591],[507,18409,18410,18413],{"style":2906},[507,18411],{"className":18412,"style":2600},[2599],[507,18414,18416],{"className":18415},[2604,2605,2606,2607],[507,18417,18419,18422,18425],{"className":18418},[2570,2607],[507,18420,2691],{"className":18421},[2570,2607],[507,18423,3293],{"className":18424},[2570,2611,2607],[507,18426,15116],{"className":18427},[2570,2611,2607],[507,18429],{"className":18430,"style":2965},[2714],[507,18432,18434,18437,18443],{"className":18433},[2937],[507,18435,4425],{"className":18436},[2941],[507,18438,18440],{"className":18439},[2570],[507,18441,3793],{"className":1844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   \\langle Y\\otimes P\\rangle&=\\frac{1}{4}\n    \\Big(\n    \\left(e^{-i\\phi}\\bra{\\psi}+i\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}-iU\\ket{\\psi}\\right)\n    \\\\\n    &\\qquad-\\left(e^{-i\\phi}\\bra{\\psi}-i\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}+iU\\ket{\\psi}\\right)\n    \\Big)\\\\\n    &=\\text{Im}\\left[e^{-i\\phi}\\bra{\\psi}PU\\ket{\\psi}\\right].\n\\end{split}\n\\end{equation}",[507,19390,19392,20271],{"className":19391,"ariaHidden":2557},[2556],[507,19393,19395,19398],{"className":19394},[2561],[507,19396],{"className":19397,"style":18174},[2565],[507,19399,19401],{"className":19400},[9452],[507,19402,19404],{"className":19403},[9711],[507,19405,19407,20263],{"className":19406},[2583,3200],[507,19408,19410,20260],{"className":19409},[2587],[507,19411,19413],{"className":19412,"style":18190},[2591],[507,19414,19415,19418],{"style":18193},[507,19416],{"className":19417,"style":18197},[2599],[507,19419,19421],{"className":19420},[2570],[507,19422,19424],{"className":19423},[2570],[507,19425,19427,19495],{"className":19426},[9452],[507,19428,19430],{"className":19429},[12327],[507,19431,19433,19487],{"className":19432},[2583,3200],[507,19434,19436,19484],{"className":19435},[2587],[507,19437,19439,19468,19476],{"className":19438,"style":18190},[2591],[507,19440,19441,19444],{"style":18193},[507,19442],{"className":19443,"style":9728},[2599],[507,19445,19447,19450,19453,19456,19459,19462,19465],{"className":19446},[2570],[507,19448,4425],{"className":19449},[2941],[507,19451,7746],{"className":19452,"style":2715},[2570,2611],[507,19454],{"className":19455,"style":2715},[2714],[507,19457,17889],{"className":19458},[2719],[507,19460],{"className":19461,"style":2715},[2714],[507,19463,3174],{"className":19464,"style":3220},[2570,2611],[507,19466,2756],{"className":19467},[2780],[507,19469,19470,19473],{"style":18250},[507,19471],{"className":19472,"style":9728},[2599],[507,19474],{"className":19475},[2570],[507,19477,19478,19481],{"style":18259},[507,19479],{"className":19480,"style":9728},[2599],[507,19482],{"className":19483},[2570],[507,19485,3225],{"className":19486},[3224],[507,19488,19490],{"className":19489},[2587],[507,19491,19493],{"className":19492,"style":18275},[2591],[507,19494],{},[507,19496,19498],{"className":19497},[12388],[507,19499,19501,20252],{"className":19500},[2583,3200],[507,19502,19504,20249],{"className":19503},[2587],[507,19505,19507,19841,20116],{"className":19506,"style":18190},[2591],[507,19508,19509,19512],{"style":18193},[507,19510],{"className":19511,"style":9728},[2599],[507,19513,19515,19518,19521,19524,19527,19589,19595,19598,19731,19734,19737,19740],{"className":19514},[2570],[507,19516],{"className":19517},[2570],[507,19519],{"className":19520,"style":2919},[2714],[507,19522,573],{"className":19523},[2923],[507,19525],{"className":19526,"style":2919},[2714],[507,19528,19530,19533,19586],{"className":19529},[2570],[507,19531],{"className":19532},[2941,9793],[507,19534,19536],{"className":19535},[9491],[507,19537,19539,19578],{"className":19538},[2583,3200],[507,19540,19542,19575],{"className":19541},[2587],[507,19543,19545,19556,19564],{"className":19544,"style":9806},[2591],[507,19546,19547,19550],{"style":11717},[507,19548],{"className":19549,"style":4310},[2599],[507,19551,19553],{"className":19552},[2570],[507,19554,12152],{"className":19555},[2570],[507,19557,19558,19561],{"style":9878},[507,19559],{"className":19560,"style":4310},[2599],[507,19562],{"className":19563,"style":9886},[9885],[507,19565,19566,19569],{"style":9889},[507,19567],{"className":19568,"style":4310},[2599],[507,19570,19572],{"className":19571},[2570],[507,19573,625],{"className":19574},[2570],[507,19576,3225],{"className":19577},[3224],[507,19579,19581],{"className":19580},[2587],[507,19582,19584],{"className":19583,"style":11755},[2591],[507,19585],{},[507,19587],{"className":19588},[2780,9793],[507,19590,19592],{"className":19591},[2570],[507,19593,580],{"className":19594},[2947,9920],[507,19596],{"className":19597,"style":2965},[2714],[507,19599,19601,19607,19645,19648,19663,19666,19669,19672,19675,19678,19693,19696,19725],{"className":19600},[2937],[507,19602,19604],{"className":19603,"style":2943},[2941,2942],[507,19605,580],{"className":19606},[2947,2948],[507,19608,19610,19613],{"className":19609},[2570],[507,19611,3286],{"className":19612},[2570,2611],[507,19614,19616],{"className":19615},[2579],[507,19617,19619],{"className":19618},[2583],[507,19620,19622],{"className":19621},[2587],[507,19623,19625],{"className":19624,"style":5040},[2591],[507,19626,19627,19630],{"style":2906},[507,19628],{"className":19629,"style":2600},[2599],[507,19631,19633],{"className":19632},[2604,2605,2606,2607],[507,19634,19636,19639,19642],{"className":19635},[2570,2607],[507,19637,2691],{"className":19638},[2570,2607],[507,19640,3293],{"className":19641},[2570,2611,2607],[507,19643,15116],{"className":19644},[2570,2611,2607],[507,19646],{"className":19647,"style":2965},[2714],[507,19649,19651,19654,19660],{"className":19650},[2937],[507,19652,4425],{"className":19653},[2941],[507,19655,19657],{"className":19656},[2570],[507,19658,3793],{"className":19659,"style":2776},[2570,2611],[507,19661,2749],{"className":19662},[2570],[507,19664],{"className":19665,"style":2715},[2714],[507,19667,2107],{"className":19668},[2719],[507,19670],{"className":19671,"style":2715},[2714],[507,19673,3293],{"className":19674},[2570,2611],[507,19676],{"className":19677,"style":2965},[2714],[507,19679,19681,19684,19690],{"className":19680},[2937],[507,19682,4425],{"className":19683},[2941],[507,19685,19687],{"className":19686},[2570],[507,19688,3793],{"className":19689,"style":2776},[2570,2611],[507,19691,2749],{"className":19692},[2570],[507,19694],{"className":19695,"style":2965},[2714],[507,19697,19699,19702],{"className":19698},[2570],[507,19700,3279],{"className":19701,"style":3314},[2570,2611],[507,19703,19705],{"className":19704},[2579],[507,19706,19708],{"className":19707},[2583],[507,19709,19711],{"className":19710},[2587],[507,19712,19714],{"className":19713,"style":5040},[2591],[507,19715,19716,19719],{"style":2906},[507,19717],{"className":19718,"style":2600},[2599],[507,19720,19722],{"className":19721},[2604,2605,2606,2607],[507,19723,17952],{"className":19724},[2719,2607],[507,19726,19728],{"className":19727,"style":2943},[2780,2942],[507,19729,3649],{"className":19730},[2947,2948],[507,19732],{"className":19733,"style":2965},[2714],[507,19735,3174],{"className":19736,"style":3220},[2570,2611],[507,19738],{"className":19739,"style":2965},[2714],[507,19741,19743,19749,19784,19787,19802,19805,19808,19811,19814,19817,19820,19835],{"className":19742},[2937],[507,19744,19746],{"className":19745,"style":2943},[2941,2942],[507,19747,580],{"className":19748},[2947,2948],[507,19750,19752,19755],{"className":19751},[2570],[507,19753,3286],{"className":19754},[2570,2611],[507,19756,19758],{"className":19757},[2579],[507,19759,19761],{"className":19760},[2583],[507,19762,19764],{"className":19763},[2587],[507,19765,19767],{"className":19766,"style":5040},[2591],[507,19768,19769,19772],{"style":2906},[507,19770],{"className":19771,"style":2600},[2599],[507,19773,19775],{"className":19774},[2604,2605,2606,2607],[507,19776,19778,19781],{"className":19777},[2570,2607],[507,19779,3293],{"className":19780},[2570,2611,2607],[507,19782,15116],{"className":19783},[2570,2611,2607],[507,19785],{"className":19786,"style":2965},[2714],[507,19788,19790,19793,19799],{"className":19789},[2937],[507,19791,2749],{"className":19792},[2570],[507,19794,19796],{"className":19795},[2570],[507,19797,3793],{"className":19798,"style":2776},[2570,2611],[507,19800,2756],{"className":19801},[2780],[507,19803],{"className":19804,"style":2715},[2714],[507,19806,2691],{"className":19807},[2719],[507,19809],{"className":19810,"style":2715},[2714],[507,19812,3293],{"className":19813},[2570,2611],[507,19815,3279],{"className":19816,"style":3314},[2570,2611],[507,19818],{"className":19819,"style":2965},[2714],[507,19821,19823,19826,19832],{"className":19822},[2937],[507,19824,2749],{"className":19825},[2570],[507,19827,19829],{"className":19828},[2570],[507,19830,3793],{"className":19831,"style":2776},[2570,2611],[507,19833,2756],{"className":19834},[2780],[507,19836,19838],{"className":19837,"style":2943},[2780,2942],[507,19839,3649],{"className":19840},[2947,2948],[507,19842,19843,19846],{"style":18250},[507,19844],{"className":19845,"style":9728},[2599],[507,19847,19849,19852,19855,19858,19861,19864,19997,20000,20003,20006,20107,20110],{"className":19848},[2570],[507,19850],{"className":19851},[2570],[507,19853],{"className":19854,"style":18629},[2714],[507,19856],{"className":19857,"style":2715},[2714],[507,19859,2691],{"className":19860},[2719],[507,19862],{"className":19863,"style":2715},[2714],[507,19865,19867,19873,19911,19914,19929,19932,19935,19938,19941,19944,19959,19962,19991],{"className":19866},[2937],[507,19868,19870],{"className":19869,"style":2943},[2941,2942],[507,19871,580],{"className":19872},[2947,2948],[507,19874,19876,19879],{"className":19875},[2570],[507,19877,3286],{"className":19878},[2570,2611],[507,19880,19882],{"className":19881},[2579],[507,19883,19885],{"className":19884},[2583],[507,19886,19888],{"className":19887},[2587],[507,19889,19891],{"className":19890,"style":5040},[2591],[507,19892,19893,19896],{"style":2906},[507,19894],{"className":19895,"style":2600},[2599],[507,19897,19899],{"className":19898},[2604,2605,2606,2607],[507,19900,19902,19905,19908],{"className":19901},[2570,2607],[507,19903,2691],{"className":19904},[2570,2607],[507,19906,3293],{"className":19907},[2570,2611,2607],[507,19909,15116],{"className":19910},[2570,2611,2607],[507,19912],{"className":19913,"style":2965},[2714],[507,19915,19917,19920,19926],{"className":19916},[2937],[507,19918,4425],{"className":19919},[2941],[507,19921,19923],{"className":19922},[2570],[507,19924,3793],{"className":19925,"style":2776},[2570,2611],[507,19927,2749],{"className":19928},[2570],[507,19930],{"className":19931,"style":2715},[2714],[507,19933,2691],{"className":19934},[2719],[507,19936],{"className":19937,"style":2715},[2714],[507,19939,3293],{"className":19940},[2570,2611],[507,19942],{"className":19943,"style":2965},[2714],[507,19945,19947,19950,19956],{"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these assumptions we were able to write the expectation values of operators of interest with fewer controlled operations. In fact, we only need to implement the controlled state preparation ",[507,20306,20308,20325],{"className":20307},[2523],[507,20309,20311],{"className":20310},[2527],[2529,20312,20313],{"xmlns":2531},[2533,20314,20315,20323],{},[2536,20316,20317,20319,20321],{},[6167,20318,7013],{},[6167,20320,7016],{},[2542,20322,3793],{},[2549,20324,14487],{"encoding":2551},[507,20326,20328],{"className":20327,"ariaHidden":2557},[2556],[507,20329,20331,20334,20340,20343],{"className":20330},[2561],[507,20332],{"className":20333,"style":7035},[2565],[507,20335,20337],{"className":20336},[2570,7039],[507,20338,7013],{"className":20339},[2570],[507,20341],{"className":20342,"style":2919},[2714],[507,20344,3793],{"className":20345,"style":2776},[2570,2611]," and not controlled time evolutions. Reframing our calculation as above will allow us to greatly reduce the depth of the resulting circuits.",[2513,20348,20350],{"id":20349},"decompose-time-evolution-operator-with-trotter-decomposition","Decompose time-evolution operator with Trotter decomposition",[18,20352,20353,20354,622,20435,622,20516,20597,20598,20627,20628,20752,20753,20810,20811,20844,20845,20894,20895,20898],{},"Instead of implementing the time-evolution operator exactly we can use the Trotter decomposition to implement an approximation of it. Repeating several times a certain order Trotter decomposition gives us further reduction of the error introduced from the approximation. In the following, we directly build the Trotter implementation in the most efficient way for the interaction graph of the Hamiltonian we are considering (nearest neighbor interactions only). In practice we insert Pauli rotations ",[507,20355,20357,20380],{"className":20356},[2523],[507,20358,20360],{"className":20359},[2527],[2529,20361,20362],{"xmlns":2531},[2533,20363,20364,20377],{},[2536,20365,20366],{},[3168,20367,20368,20371],{},[2542,20369,20370],{},"R",[2536,20372,20373,20375],{},[2542,20374,9139],{},[2542,20376,9139],{},[2549,20378,20379],{"encoding":2551},"R_{xx}",[507,20381,20383],{"className":20382,"ariaHidden":2557},[2556],[507,20384,20386,20389],{"className":20385},[2561],[507,20387],{"className":20388,"style":3187},[2565],[507,20390,20392,20396],{"className":20391},[2570],[507,20393,20370],{"className":20394,"style":20395},[2570,2611],"margin-right:0.0077em;",[507,20397,20399],{"className":20398},[2579],[507,20400,20402,20427],{"className":20401},[2583,3200],[507,20403,20405,20424],{"className":20404},[2587],[507,20406,20408],{"className":20407,"style":4507},[2591],[507,20409,20411,20414],{"style":20410},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[507,20412],{"className":20413,"style":2600},[2599],[507,20415,20417],{"className":20416},[2604,2605,2606,2607],[507,20418,20420],{"className":20419},[2570,2607],[507,20421,20423],{"className":20422},[2570,2611,2607],"xx",[507,20425,3225],{"className":20426},[3224],[507,20428,20430],{"className":20429},[2587],[507,20431,20433],{"className":20432,"style":3232},[2591],[507,20434],{},[507,20436,20438,20461],{"className":20437},[2523],[507,20439,20441],{"className":20440},[2527],[2529,20442,20443],{"xmlns":2531},[2533,20444,20445,20458],{},[2536,20446,20447],{},[3168,20448,20449,20451],{},[2542,20450,20370],{},[2536,20452,20453,20456],{},[2542,20454,20455],{},"y",[2542,20457,20455],{},[2549,20459,20460],{"encoding":2551},"R_{yy}",[507,20462,20464],{"className":20463,"ariaHidden":2557},[2556],[507,20465,20467,20470],{"className":20466},[2561],[507,20468],{"className":20469,"style":7967},[2565],[507,20471,20473,20476],{"className":20472},[2570],[507,20474,20370],{"className":20475,"style":20395},[2570,2611],[507,20477,20479],{"className":20478},[2579],[507,20480,20482,20508],{"className":20481},[2583,3200],[507,20483,20485,20505],{"className":20484},[2587],[507,20486,20488],{"className":20487,"style":4507},[2591],[507,20489,20490,20493],{"style":20410},[507,20491],{"className":20492,"style":2600},[2599],[507,20494,20496],{"className":20495},[2604,2605,2606,2607],[507,20497,20499,20502],{"className":20498},[2570,2607],[507,20500,20455],{"className":20501,"style":2776},[2570,2611,2607],[507,20503,20455],{"className":20504,"style":2776},[2570,2611,2607],[507,20506,3225],{"className":20507},[3224],[507,20509,20511],{"className":20510},[2587],[507,20512,20514],{"className":20513,"style":6833},[2591],[507,20515],{},[507,20517,20519,20541],{"className":20518},[2523],[507,20520,20522],{"className":20521},[2527],[2529,20523,20524],{"xmlns":2531},[2533,20525,20526,20538],{},[2536,20527,20528],{},[3168,20529,20530,20532],{},[2542,20531,20370],{},[2536,20533,20534,20536],{},[2542,20535,666],{},[2542,20537,666],{},[2549,20539,20540],{"encoding":2551},"R_{zz}",[507,20542,20544],{"className":20543,"ariaHidden":2557},[2556],[507,20545,20547,20550],{"className":20546},[2561],[507,20548],{"className":20549,"style":3187},[2565],[507,20551,20553,20556],{"className":20552},[2570],[507,20554,20370],{"className":20555,"style":20395},[2570,2611],[507,20557,20559],{"className":20558},[2579],[507,20560,20562,20589],{"className":20561},[2583,3200],[507,20563,20565,20586],{"className":20564},[2587],[507,20566,20568],{"className":20567,"style":4507},[2591],[507,20569,20570,20573],{"style":20410},[507,20571],{"className":20572,"style":2600},[2599],[507,20574,20576],{"className":20575},[2604,2605,2606,2607],[507,20577,20579,20583],{"className":20578},[2570,2607],[507,20580,666],{"className":20581,"style":20582},[2570,2611,2607],"margin-right:0.044em;",[507,20584,666],{"className":20585,"style":20582},[2570,2611,2607],[507,20587,3225],{"className":20588},[3224],[507,20590,20592],{"className":20591},[2587],[507,20593,20595],{"className":20594,"style":3232},[2591],[507,20596],{}," with a parametrized angle ",[507,20599,20601,20614],{"className":20600},[2523],[507,20602,20604],{"className":20603},[2527],[2529,20605,20606],{"xmlns":2531},[2533,20607,20608,20612],{},[2536,20609,20610],{},[2542,20611,3298],{},[2549,20613,3298],{"encoding":2551},[507,20615,20617],{"className":20616,"ariaHidden":2557},[2556],[507,20618,20620,20624],{"className":20619},[2561],[507,20621],{"className":20622,"style":20623},[2565],"height:0.6151em;",[507,20625,3298],{"className":20626},[2570,2611]," which correspond to the approximate implementation of ",[507,20629,20631,20675],{"className":20630},[2523],[507,20632,20634],{"className":20633},[2527],[2529,20635,20636],{"xmlns":2531},[2533,20637,20638,20672],{},[2536,20639,20640],{},[2539,20641,20642,20644],{},[2542,20643,3286],{},[2536,20645,20646,20648,20650,20652,20654,20656,20658,20660,20662,20664,20666,20668,20670],{},[2689,20647,2691],{},[2542,20649,3293],{},[2689,20651,580],{"stretchy":2755},[2542,20653,7731],{},[2542,20655,7731],{},[2689,20657,2107],{},[2542,20659,7746],{},[2542,20661,7746],{},[2689,20663,2107],{},[2542,20665,7764],{},[2542,20667,7764],{},[2689,20669,3649],{"stretchy":2755},[2542,20671,3298],{},[2549,20673,20674],{"encoding":2551},"e^{-i (XX + YY + ZZ) t}",[507,20676,20678],{"className":20677,"ariaHidden":2557},[2556],[507,20679,20681,20684],{"className":20680},[2561],[507,20682],{"className":20683,"style":3708},[2565],[507,20685,20687,20690],{"className":20686},[2570],[507,20688,3286],{"className":20689},[2570,2611],[507,20691,20693],{"className":20692},[2579],[507,20694,20696],{"className":20695},[2583],[507,20697,20699],{"className":20698},[2587],[507,20700,20702],{"className":20701,"style":3708},[2591],[507,20703,20704,20707],{"style":2595},[507,20705],{"className":20706,"style":2600},[2599],[507,20708,20710],{"className":20709},[2604,2605,2606,2607],[507,20711,20713,20716,20719,20722,20725,20728,20731,20734,20737,20740,20743,20746,20749],{"className":20712},[2570,2607],[507,20714,2691],{"className":20715},[2570,2607],[507,20717,3293],{"className":20718},[2570,2611,2607],[507,20720,580],{"className":20721},[2941,2607],[507,20723,7731],{"className":20724,"style":7876},[2570,2611,2607],[507,20726,7731],{"className":20727,"style":7876},[2570,2611,2607],[507,20729,2107],{"className":20730},[2719,2607],[507,20732,7746],{"className":20733,"style":2715},[2570,2611,2607],[507,20735,7746],{"className":20736,"style":2715},[2570,2611,2607],[507,20738,2107],{"className":20739},[2719,2607],[507,20741,7764],{"className":20742,"style":8074},[2570,2611,2607],[507,20744,7764],{"className":20745,"style":8074},[2570,2611,2607],[507,20747,3649],{"className":20748},[2780,2607],[507,20750,3298],{"className":20751},[2570,2611,2607],". Given the difference in definition of the Pauli rotations and the time-evolution that we are trying to implement, we'll have to use the parameter ",[507,20754,20756,20777],{"className":20755},[2523],[507,20757,20759],{"className":20758},[2527],[2529,20760,20761],{"xmlns":2531},[2533,20762,20763,20774],{},[2536,20764,20765,20767,20770,20772],{},[2693,20766,584],{},[2689,20768,20769],{},"∗",[2542,20771,4959],{},[2542,20773,3298],{},[2549,20775,20776],{"encoding":2551},"2*dt",[507,20778,20780,20798],{"className":20779,"ariaHidden":2557},[2556],[507,20781,20783,20786,20789,20792,20795],{"className":20782},[2561],[507,20784],{"className":20785,"style":2729},[2565],[507,20787,584],{"className":20788},[2570],[507,20790],{"className":20791,"style":2715},[2714],[507,20793,20769],{"className":20794},[2719],[507,20796],{"className":20797,"style":2715},[2714],[507,20799,20801,20804,20807],{"className":20800},[2561],[507,20802],{"className":20803,"style":5434},[2565],[507,20805,4959],{"className":20806},[2570,2611],[507,20808,3298],{"className":20809},[2570,2611]," to achieve a time-evolution of ",[507,20812,20814,20829],{"className":20813},[2523],[507,20815,20817],{"className":20816},[2527],[2529,20818,20819],{"xmlns":2531},[2533,20820,20821,20827],{},[2536,20822,20823,20825],{},[2542,20824,4959],{},[2542,20826,3298],{},[2549,20828,5463],{"encoding":2551},[507,20830,20832],{"className":20831,"ariaHidden":2557},[2556],[507,20833,20835,20838,20841],{"className":20834},[2561],[507,20836],{"className":20837,"style":5434},[2565],[507,20839,4959],{"className":20840},[2570,2611],[507,20842,3298],{"className":20843},[2570,2611],". Furthermore, we reverse the order of the operations for odd number of repetitions of the Trotter steps, which is functionally equivalent but allows for synthesizing adjacent operations in a single ",[507,20846,20848,20870],{"className":20847},[2523],[507,20849,20851],{"className":20850},[2527],[2529,20852,20853],{"xmlns":2531},[2533,20854,20855,20867],{},[2536,20856,20857,20859,20861,20863,20865],{},[2542,20858,5857],{},[2542,20860,3279],{},[2689,20862,580],{"stretchy":2755},[2693,20864,584],{},[2689,20866,3649],{"stretchy":2755},[2549,20868,20869],{"encoding":2551},"SU(2)",[507,20871,20873],{"className":20872,"ariaHidden":2557},[2556],[507,20874,20876,20879,20882,20885,20888,20891],{"className":20875},[2561],[507,20877],{"className":20878,"style":2769},[2565],[507,20880,5857],{"className":20881,"style":5892},[2570,2611],[507,20883,3279],{"className":20884,"style":3314},[2570,2611],[507,20886,580],{"className":20887},[2941],[507,20889,584],{"className":20890},[2570],[507,20892,3649],{"className":20893},[2780]," unitary. This gives a much shallower circuit than what is obtained using the generic ",[504,20896,20897],{},"PauliEvolutionGate()"," functionality.",[498,20900,20902],{"className":500,"code":20901,"language":502,"meta":104,"style":104},"t = Parameter(\"t\")\n\n# Create instruction for rotation about XX+YY-ZZ:\nRxyz_circ = QuantumCircuit(2)\nRxyz_circ.rxx(t, 0, 1)\nRxyz_circ.ryy(t, 0, 1)\nRxyz_circ.rzz(t, 0, 1)\nRxyz_instr = Rxyz_circ.to_instruction(label=\"RXX+YY+ZZ\")\n\ninteraction_list = [\n    [[i, i + 1] for i in range(0, n_qubits - 1, 2)],\n    [[i, i + 1] for i in range(1, n_qubits - 1, 2)],\n]  # linear chain\n\nqr = QuantumRegister(n_qubits)\ntrotter_step_circ = QuantumCircuit(qr)\nfor i, color in enumerate(interaction_list):\n    for interaction in color:\n        trotter_step_circ.append(Rxyz_instr, interaction)\n    if i \u003C len(interaction_list) - 1:\n        trotter_step_circ.barrier()\nreverse_trotter_step_circ = trotter_step_circ.reverse_ops()\n\nqc_evol = QuantumCircuit(qr)\nfor step in range(num_trotter_steps):\n    if step % 2 == 0:\n        qc_evol = qc_evol.compose(trotter_step_circ)\n    else:\n        qc_evol = qc_evol.compose(reverse_trotter_step_circ)\n\nqc_evol.decompose().draw(\"mpl\", fold=-1, scale=0.5)\n",[504,20903,20904,20918,20922,20927,20942,20961,20978,20995,21020,21024,21033,21070,21104,21112,21116,21126,21137,21151,21163,21173,21192,21201,21216,21220,21230,21243,21261,21276,21282,21295,21299],{"__ignoreMap":104},[507,20905,20906,20908,20910,20912,20914,20916],{"class":509,"line":510},[507,20907,9307],{"class":517},[507,20909,573],{"class":572},[507,20911,9312],{"class":576},[507,20913,580],{"class":517},[507,20915,9317],{"class":730},[507,20917,587],{"class":517},[507,20919,20920],{"class":509,"line":105},[507,20921,556],{"emptyLinePlaceholder":133},[507,20923,20924],{"class":509,"line":540},[507,20925,20926],{"class":562},"# Create instruction for rotation about XX+YY-ZZ:\n",[507,20928,20929,20932,20934,20936,20938,20940],{"class":509,"line":553},[507,20930,20931],{"class":517},"Rxyz_circ ",[507,20933,573],{"class":572},[507,20935,577],{"class":576},[507,20937,580],{"class":517},[507,20939,584],{"class":583},[507,20941,587],{"class":517},[507,20943,20944,20947,20950,20953,20955,20957,20959],{"class":509,"line":559},[507,20945,20946],{"class":517},"Rxyz_circ.",[507,20948,20949],{"class":576},"rxx",[507,20951,20952],{"class":517},"(t, ",[507,20954,601],{"class":583},[507,20956,622],{"class":517},[507,20958,625],{"class":583},[507,20960,587],{"class":517},[507,20962,20963,20965,20968,20970,20972,20974,20976],{"class":509,"line":566},[507,20964,20946],{"class":517},[507,20966,20967],{"class":576},"ryy",[507,20969,20952],{"class":517},[507,20971,601],{"class":583},[507,20973,622],{"class":517},[507,20975,625],{"class":583},[507,20977,587],{"class":517},[507,20979,20980,20982,20985,20987,20989,20991,20993],{"class":509,"line":590},[507,20981,20946],{"class":517},[507,20983,20984],{"class":576},"rzz",[507,20986,20952],{"class":517},[507,20988,601],{"class":583},[507,20990,622],{"class":517},[507,20992,625],{"class":583},[507,20994,587],{"class":517},[507,20996,20997,21000,21002,21005,21008,21010,21013,21015,21018],{"class":509,"line":610},[507,20998,20999],{"class":517},"Rxyz_instr ",[507,21001,573],{"class":572},[507,21003,21004],{"class":517}," Rxyz_circ.",[507,21006,21007],{"class":576},"to_instruction",[507,21009,580],{"class":517},[507,21011,21012],{"class":2155},"label",[507,21014,573],{"class":572},[507,21016,21017],{"class":730},"\"RXX+YY+ZZ\"",[507,21019,587],{"class":517},[507,21021,21022],{"class":509,"line":634},[507,21023,556],{"emptyLinePlaceholder":133},[507,21025,21026,21029,21031],{"class":509,"line":661},[507,21027,21028],{"class":517},"interaction_list ",[507,21030,573],{"class":572},[507,21032,2177],{"class":517},[507,21034,21035,21038,21040,21042,21044,21046,21048,21050,21052,21054,21056,21059,21061,21063,21065,21067],{"class":509,"line":678},[507,21036,21037],{"class":517},"    [[i, i ",[507,21039,2107],{"class":572},[507,21041,1426],{"class":583},[507,21043,8206],{"class":517},[507,21045,1630],{"class":513},[507,21047,8246],{"class":517},[507,21049,1636],{"class":513},[507,21051,8221],{"class":572},[507,21053,580],{"class":517},[507,21055,601],{"class":583},[507,21057,21058],{"class":517},", n_qubits ",[507,21060,2367],{"class":572},[507,21062,1426],{"class":583},[507,21064,622],{"class":517},[507,21066,584],{"class":583},[507,21068,21069],{"class":517},")],\n",[507,21071,21072,21074,21076,21078,21080,21082,21084,21086,21088,21090,21092,21094,21096,21098,21100,21102],{"class":509,"line":683},[507,21073,21037],{"class":517},[507,21075,2107],{"class":572},[507,21077,1426],{"class":583},[507,21079,8206],{"class":517},[507,21081,1630],{"class":513},[507,21083,8246],{"class":517},[507,21085,1636],{"class":513},[507,21087,8221],{"class":572},[507,21089,580],{"class":517},[507,21091,625],{"class":583},[507,21093,21058],{"class":517},[507,21095,2367],{"class":572},[507,21097,1426],{"class":583},[507,21099,622],{"class":517},[507,21101,584],{"class":583},[507,21103,21069],{"class":517},[507,21105,21106,21109],{"class":509,"line":697},[507,21107,21108],{"class":517},"]  ",[507,21110,21111],{"class":562},"# linear chain\n",[507,21113,21114],{"class":509,"line":710},[507,21115,556],{"emptyLinePlaceholder":133},[507,21117,21118,21120,21122,21124],{"class":509,"line":715},[507,21119,9384],{"class":517},[507,21121,573],{"class":572},[507,21123,9389],{"class":576},[507,21125,8783],{"class":517},[507,21127,21128,21131,21133,21135],{"class":509,"line":721},[507,21129,21130],{"class":517},"trotter_step_circ ",[507,21132,573],{"class":572},[507,21134,577],{"class":576},[507,21136,9403],{"class":517},[507,21138,21139,21141,21144,21146,21148],{"class":509,"line":736},[507,21140,1630],{"class":513},[507,21142,21143],{"class":517}," i, color ",[507,21145,1636],{"class":513},[507,21147,1957],{"class":572},[507,21149,21150],{"class":517},"(interaction_list):\n",[507,21152,21153,21155,21158,21160],{"class":509,"line":748},[507,21154,1916],{"class":513},[507,21156,21157],{"class":517}," interaction ",[507,21159,1636],{"class":513},[507,21161,21162],{"class":517}," color:\n",[507,21164,21165,21168,21170],{"class":509,"line":761},[507,21166,21167],{"class":517},"        trotter_step_circ.",[507,21169,1939],{"class":576},[507,21171,21172],{"class":517},"(Rxyz_instr, interaction)\n",[507,21174,21175,21177,21179,21181,21183,21186,21188,21190],{"class":509,"line":775},[507,21176,1717],{"class":513},[507,21178,8246],{"class":517},[507,21180,5677],{"class":572},[507,21182,2301],{"class":572},[507,21184,21185],{"class":517},"(interaction_list) ",[507,21187,2367],{"class":572},[507,21189,1426],{"class":583},[507,21191,1728],{"class":517},[507,21193,21194,21196,21199],{"class":509,"line":784},[507,21195,21167],{"class":517},[507,21197,21198],{"class":576},"barrier",[507,21200,781],{"class":517},[507,21202,21203,21206,21208,21211,21214],{"class":509,"line":796},[507,21204,21205],{"class":517},"reverse_trotter_step_circ ",[507,21207,573],{"class":572},[507,21209,21210],{"class":517}," trotter_step_circ.",[507,21212,21213],{"class":576},"reverse_ops",[507,21215,781],{"class":517},[507,21217,21218],{"class":509,"line":809},[507,21219,556],{"emptyLinePlaceholder":133},[507,21221,21222,21224,21226,21228],{"class":509,"line":1352},[507,21223,9396],{"class":517},[507,21225,573],{"class":572},[507,21227,577],{"class":576},[507,21229,9403],{"class":517},[507,21231,21232,21234,21237,21239,21241],{"class":509,"line":1357},[507,21233,1630],{"class":513},[507,21235,21236],{"class":517}," step ",[507,21238,1636],{"class":513},[507,21240,8221],{"class":572},[507,21242,13893],{"class":517},[507,21244,21245,21247,21249,21251,21253,21256,21259],{"class":509,"line":1362},[507,21246,1717],{"class":513},[507,21248,21236],{"class":517},[507,21250,8769],{"class":572},[507,21252,2316],{"class":583},[507,21254,21255],{"class":572}," ==",[507,21257,21258],{"class":583}," 0",[507,21260,1728],{"class":517},[507,21262,21263,21266,21268,21271,21273],{"class":509,"line":1367},[507,21264,21265],{"class":517},"        qc_evol ",[507,21267,573],{"class":572},[507,21269,21270],{"class":517}," qc_evol.",[507,21272,13867],{"class":576},[507,21274,21275],{"class":517},"(trotter_step_circ)\n",[507,21277,21278,21280],{"class":509,"line":1379},[507,21279,1800],{"class":513},[507,21281,1728],{"class":517},[507,21283,21284,21286,21288,21290,21292],{"class":509,"line":1389},[507,21285,21265],{"class":517},[507,21287,573],{"class":572},[507,21289,21270],{"class":517},[507,21291,13867],{"class":576},[507,21293,21294],{"class":517},"(reverse_trotter_step_circ)\n",[507,21296,21297],{"class":509,"line":1397},[507,21298,556],{"emptyLinePlaceholder":133},[507,21300,21301,21303,21305,21307,21309,21311,21313,21315,21317,21319,21321,21323,21325,21327,21329],{"class":509,"line":1412},[507,21302,9408],{"class":517},[507,21304,14165],{"class":576},[507,21306,13983],{"class":517},[507,21308,9164],{"class":576},[507,21310,580],{"class":517},[507,21312,9169],{"class":730},[507,21314,622],{"class":517},[507,21316,14112],{"class":2155},[507,21318,14115],{"class":572},[507,21320,625],{"class":583},[507,21322,622],{"class":517},[507,21324,9174],{"class":2155},[507,21326,573],{"class":572},[507,21328,9179],{"class":583},[507,21330,587],{"class":517},[831,21332],{"alt":9184,"src":21333},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-03.avif",[2513,21335,21337],{"id":21336},"use-an-optimized-circuit-for-state-preparation","Use an optimized circuit for state preparation",[498,21339,21341],{"className":500,"code":21340,"language":502,"meta":104,"style":104},"control = 0\nexcitation = int(n_qubits \u002F 2) + 1\ncontrolled_state_prep = QuantumCircuit(n_qubits + 1)\ncontrolled_state_prep.cx(control, excitation)\ncontrolled_state_prep.draw(\"mpl\", fold=-1, scale=0.5)\n",[504,21342,21343,21352,21373,21390,21400],{"__ignoreMap":104},[507,21344,21345,21348,21350],{"class":509,"line":510},[507,21346,21347],{"class":517},"control ",[507,21349,573],{"class":572},[507,21351,2246],{"class":583},[507,21353,21354,21357,21359,21361,21363,21365,21367,21369,21371],{"class":509,"line":105},[507,21355,21356],{"class":517},"excitation ",[507,21358,573],{"class":572},[507,21360,2095],{"class":572},[507,21362,2104],{"class":517},[507,21364,645],{"class":572},[507,21366,2316],{"class":583},[507,21368,655],{"class":517},[507,21370,2107],{"class":572},[507,21372,2084],{"class":583},[507,21374,21375,21378,21380,21382,21384,21386,21388],{"class":509,"line":540},[507,21376,21377],{"class":517},"controlled_state_prep ",[507,21379,573],{"class":572},[507,21381,577],{"class":576},[507,21383,2104],{"class":517},[507,21385,2107],{"class":572},[507,21387,1426],{"class":583},[507,21389,587],{"class":517},[507,21391,21392,21395,21397],{"class":509,"line":553},[507,21393,21394],{"class":517},"controlled_state_prep.",[507,21396,615],{"class":576},[507,21398,21399],{"class":517},"(control, excitation)\n",[507,21401,21402,21404,21406,21408,21410,21412,21414,21416,21418,21420,21422,21424,21426],{"class":509,"line":559},[507,21403,21394],{"class":517},[507,21405,9164],{"class":576},[507,21407,580],{"class":517},[507,21409,9169],{"class":730},[507,21411,622],{"class":517},[507,21413,14112],{"class":2155},[507,21415,14115],{"class":572},[507,21417,625],{"class":583},[507,21419,622],{"class":517},[507,21421,9174],{"class":2155},[507,21423,573],{"class":572},[507,21425,9179],{"class":583},[507,21427,587],{"class":517},[831,21429],{"alt":9184,"src":21430},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-04.avif",[2513,21432,21434,21435,10799,21495,21555],{"id":21433},"template-circuits-for-calculating-matrix-elements-of-stildess-and-htildehh-via-hadamard-test","Template circuits for calculating matrix elements of ",[507,21436,21438,21455],{"className":21437},[2523],[507,21439,21441],{"className":21440},[2527],[2529,21442,21443],{"xmlns":2531},[2533,21444,21445,21453],{},[2536,21446,21447],{},[4271,21448,21449,21451],{"accent":2557},[2542,21450,5857],{},[2689,21452,4277],{},[2549,21454,5862],{"encoding":2551},[507,21456,21458],{"className":21457,"ariaHidden":2557},[2556],[507,21459,21461,21464],{"className":21460},[2561],[507,21462],{"className":21463,"style":4290},[2565],[507,21465,21467],{"className":21466},[2570,4294],[507,21468,21470],{"className":21469},[2583],[507,21471,21473],{"className":21472},[2587],[507,21474,21476,21484],{"className":21475,"style":4290},[2591],[507,21477,21478,21481],{"style":4306},[507,21479],{"className":21480,"style":4310},[2599],[507,21482,5857],{"className":21483,"style":5892},[2570,2611],[507,21485,21486,21489],{"style":4316},[507,21487],{"className":21488,"style":4310},[2599],[507,21490,21492],{"className":21491,"style":5901},[4323],[507,21493,4277],{"className":21494},[2570],[507,21496,21498,21515],{"className":21497},[2523],[507,21499,21501],{"className":21500},[2527],[2529,21502,21503],{"xmlns":2531},[2533,21504,21505,21513],{},[2536,21506,21507],{},[4271,21508,21509,21511],{"accent":2557},[2542,21510,3138],{},[2689,21512,4277],{},[2549,21514,4280],{"encoding":2551},[507,21516,21518],{"className":21517,"ariaHidden":2557},[2556],[507,21519,21521,21524],{"className":21520},[2561],[507,21522],{"className":21523,"style":4290},[2565],[507,21525,21527],{"className":21526},[2570,4294],[507,21528,21530],{"className":21529},[2583],[507,21531,21533],{"className":21532},[2587],[507,21534,21536,21544],{"className":21535,"style":4290},[2591],[507,21537,21538,21541],{"style":4306},[507,21539],{"className":21540,"style":4310},[2599],[507,21542,3138],{"className":21543,"style":3153},[2570,2611],[507,21545,21546,21549],{"style":4316},[507,21547],{"className":21548,"style":4310},[2599],[507,21550,21552],{"className":21551,"style":4324},[4323],[507,21553,4277],{"className":21554},[2570]," via Hadamard test",[18,21557,21558],{},"The only difference between the circuits used in the Hadamard test will be the phase in the time-evolution operator and the observables measured. Therefore we can prepare a template circuit which represent the generic circuit for the Hadamard test, with placeholders for the gates that depend on the time-evolution operator.",[498,21560,21562],{"className":500,"code":21561,"language":502,"meta":104,"style":104},"# Parameters for the template circuits\nparameters = []\nfor idx in range(1, krylov_dim):\n    parameters.append(2 * dt_circ * (idx))\n",[504,21563,21564,21569,21578,21596],{"__ignoreMap":104},[507,21565,21566],{"class":509,"line":510},[507,21567,21568],{"class":562},"# Parameters for the template circuits\n",[507,21570,21571,21574,21576],{"class":509,"line":105},[507,21572,21573],{"class":517},"parameters ",[507,21575,573],{"class":572},[507,21577,1910],{"class":517},[507,21579,21580,21582,21585,21587,21589,21591,21593],{"class":509,"line":540},[507,21581,1630],{"class":513},[507,21583,21584],{"class":517}," idx ",[507,21586,1636],{"class":513},[507,21588,8221],{"class":572},[507,21590,580],{"class":517},[507,21592,625],{"class":583},[507,21594,21595],{"class":517},", krylov_dim):\n",[507,21597,21598,21601,21603,21605,21607,21609,21612,21614],{"class":509,"line":553},[507,21599,21600],{"class":517},"    parameters.",[507,21602,1939],{"class":576},[507,21604,580],{"class":517},[507,21606,584],{"class":583},[507,21608,8229],{"class":572},[507,21610,21611],{"class":517}," dt_circ ",[507,21613,2391],{"class":572},[507,21615,21616],{"class":517}," (idx))\n",[498,21618,21620],{"className":500,"code":21619,"language":502,"meta":104,"style":104},"# Create modified hadamard test circuit\nqr = QuantumRegister(n_qubits + 1)\nqc = QuantumCircuit(qr)\nqc.h(0)\nqc.compose(controlled_state_prep, list(range(n_qubits + 1)), inplace=True)\nqc.barrier()\nqc.compose(qc_evol, list(range(1, n_qubits + 1)), inplace=True)\nqc.barrier()\nqc.x(0)\nqc.compose(\n    controlled_state_prep.inverse(), list(range(n_qubits + 1)), inplace=True\n)\nqc.x(0)\n\nqc.decompose().draw(\"mpl\", fold=-1)\n",[504,21621,21622,21627,21643,21653,21665,21697,21705,21740,21748,21760,21768,21798,21802,21814,21818],{"__ignoreMap":104},[507,21623,21624],{"class":509,"line":510},[507,21625,21626],{"class":562},"# Create modified hadamard test circuit\n",[507,21628,21629,21631,21633,21635,21637,21639,21641],{"class":509,"line":105},[507,21630,9384],{"class":517},[507,21632,573],{"class":572},[507,21634,9389],{"class":576},[507,21636,2104],{"class":517},[507,21638,2107],{"class":572},[507,21640,1426],{"class":583},[507,21642,587],{"class":517},[507,21644,21645,21647,21649,21651],{"class":509,"line":540},[507,21646,569],{"class":517},[507,21648,573],{"class":572},[507,21650,577],{"class":576},[507,21652,9403],{"class":517},[507,21654,21655,21657,21659,21661,21663],{"class":509,"line":553},[507,21656,593],{"class":517},[507,21658,596],{"class":576},[507,21660,580],{"class":517},[507,21662,601],{"class":583},[507,21664,587],{"class":517},[507,21666,21667,21669,21671,21674,21676,21678,21680,21682,21684,21686,21689,21691,21693,21695],{"class":509,"line":559},[507,21668,593],{"class":517},[507,21670,13867],{"class":576},[507,21672,21673],{"class":517},"(controlled_state_prep, ",[507,21675,14053],{"class":572},[507,21677,580],{"class":517},[507,21679,2204],{"class":572},[507,21681,2104],{"class":517},[507,21683,2107],{"class":572},[507,21685,1426],{"class":583},[507,21687,21688],{"class":517},")), ",[507,21690,13873],{"class":2155},[507,21692,573],{"class":572},[507,21694,13878],{"class":583},[507,21696,587],{"class":517},[507,21698,21699,21701,21703],{"class":509,"line":566},[507,21700,593],{"class":517},[507,21702,21198],{"class":576},[507,21704,781],{"class":517},[507,21706,21707,21709,21711,21714,21716,21718,21720,21722,21724,21726,21728,21730,21732,21734,21736,21738],{"class":509,"line":590},[507,21708,593],{"class":517},[507,21710,13867],{"class":576},[507,21712,21713],{"class":517},"(qc_evol, ",[507,21715,14053],{"class":572},[507,21717,580],{"class":517},[507,21719,2204],{"class":572},[507,21721,580],{"class":517},[507,21723,625],{"class":583},[507,21725,21058],{"class":517},[507,21727,2107],{"class":572},[507,21729,1426],{"class":583},[507,21731,21688],{"class":517},[507,21733,13873],{"class":2155},[507,21735,573],{"class":572},[507,21737,13878],{"class":583},[507,21739,587],{"class":517},[507,21741,21742,21744,21746],{"class":509,"line":610},[507,21743,593],{"class":517},[507,21745,21198],{"class":576},[507,21747,781],{"class":517},[507,21749,21750,21752,21754,21756,21758],{"class":509,"line":634},[507,21751,593],{"class":517},[507,21753,9139],{"class":576},[507,21755,580],{"class":517},[507,21757,601],{"class":583},[507,21759,587],{"class":517},[507,21761,21762,21764,21766],{"class":509,"line":661},[507,21763,593],{"class":517},[507,21765,13867],{"class":576},[507,21767,1376],{"class":517},[507,21769,21770,21773,21775,21777,21779,21781,21783,21785,21787,21789,21791,21793,21795],{"class":509,"line":678},[507,21771,21772],{"class":517},"    controlled_state_prep.",[507,21774,13825],{"class":576},[507,21776,13950],{"class":517},[507,21778,14053],{"class":572},[507,21780,580],{"class":517},[507,21782,2204],{"class":572},[507,21784,2104],{"class":517},[507,21786,2107],{"class":572},[507,21788,1426],{"class":583},[507,21790,21688],{"class":517},[507,21792,13873],{"class":2155},[507,21794,573],{"class":572},[507,21796,21797],{"class":583},"True\n",[507,21799,21800],{"class":509,"line":683},[507,21801,587],{"class":517},[507,21803,21804,21806,21808,21810,21812],{"class":509,"line":697},[507,21805,593],{"class":517},[507,21807,9139],{"class":576},[507,21809,580],{"class":517},[507,21811,601],{"class":583},[507,21813,587],{"class":517},[507,21815,21816],{"class":509,"line":710},[507,21817,556],{"emptyLinePlaceholder":133},[507,21819,21820,21822,21824,21826,21828,21830,21832,21834,21836,21838,21840],{"class":509,"line":715},[507,21821,593],{"class":517},[507,21823,14165],{"class":576},[507,21825,13983],{"class":517},[507,21827,9164],{"class":576},[507,21829,580],{"class":517},[507,21831,9169],{"class":730},[507,21833,622],{"class":517},[507,21835,14112],{"class":2155},[507,21837,14115],{"class":572},[507,21839,625],{"class":583},[507,21841,587],{"class":517},[831,21843],{"alt":9184,"src":21844},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-05.avif",[498,21846,21848],{"className":500,"code":21847,"language":502,"meta":104,"style":104},"print(\n    \"The optimized circuit has 2Q gates depth: \",\n    qc.decompose().decompose().depth(lambda x: x[0].num_qubits == 2),\n)\n",[504,21849,21850,21856,21863,21896],{"__ignoreMap":104},[507,21851,21852,21854],{"class":509,"line":510},[507,21853,8525],{"class":572},[507,21855,1376],{"class":517},[507,21857,21858,21861],{"class":509,"line":105},[507,21859,21860],{"class":730},"    \"The optimized circuit has 2Q gates depth: \"",[507,21862,1409],{"class":517},[507,21864,21865,21868,21870,21872,21874,21876,21878,21880,21882,21884,21886,21888,21890,21892,21894],{"class":509,"line":540},[507,21866,21867],{"class":517},"    qc.",[507,21869,14165],{"class":576},[507,21871,13983],{"class":517},[507,21873,14165],{"class":576},[507,21875,13983],{"class":517},[507,21877,14179],{"class":576},[507,21879,580],{"class":517},[507,21881,14184],{"class":513},[507,21883,14187],{"class":1382},[507,21885,14190],{"class":517},[507,21887,601],{"class":583},[507,21889,14195],{"class":517},[507,21891,1723],{"class":572},[507,21893,2316],{"class":583},[507,21895,14202],{"class":517},[507,21897,21898],{"class":509,"line":553},[507,21899,587],{"class":517},[498,21901,21904],{"className":21902,"code":21903,"language":7039,"meta":104},[8531],"The optimized circuit has 2Q gates depth:  74\n",[504,21905,21903],{"__ignoreMap":104},[18,21907,21908],{},"We have considerably reduced the depth of the Hadamard test with a combination of Trotter approximation and uncontrolled unitaries",[13,21910,21912],{"id":21911},"step-3-execute-using-qiskit-primitives","Step 3: Execute using Qiskit primitives",[18,21914,21915],{},"Instantiate the backend and set runtime parameters",[498,21917,21919],{"className":500,"code":21918,"language":502,"meta":104,"style":104},"service = QiskitRuntimeService()\nbackend = service.least_busy(operational=True, simulator=False)\nif (\n    \"if_else\" not in backend.target.operation_names\n):  # Needed as \"op_name\" could be \"if_else\"\n    backend.target.add_instruction(IfElseOp, name=\"if_else\")\nprint(backend.name)\n",[504,21920,21921,21933,21967,21973,21987,21995,22016],{"__ignoreMap":104},[507,21922,21923,21926,21928,21931],{"class":509,"line":510},[507,21924,21925],{"class":517},"service ",[507,21927,573],{"class":572},[507,21929,21930],{"class":576}," QiskitRuntimeService",[507,21932,781],{"class":517},[507,21934,21935,21938,21940,21943,21946,21948,21951,21953,21955,21957,21960,21962,21965],{"class":509,"line":105},[507,21936,21937],{"class":517},"backend ",[507,21939,573],{"class":572},[507,21941,21942],{"class":517}," service.",[507,21944,21945],{"class":576},"least_busy",[507,21947,580],{"class":517},[507,21949,21950],{"class":2155},"operational",[507,21952,573],{"class":572},[507,21954,13878],{"class":583},[507,21956,622],{"class":517},[507,21958,21959],{"class":2155},"simulator",[507,21961,573],{"class":572},[507,21963,21964],{"class":583},"False",[507,21966,587],{"class":517},[507,21968,21969,21971],{"class":509,"line":540},[507,21970,1645],{"class":513},[507,21972,1334],{"class":517},[507,21974,21975,21978,21981,21984],{"class":509,"line":553},[507,21976,21977],{"class":730},"    \"if_else\"",[507,21979,21980],{"class":513}," not",[507,21982,21983],{"class":513}," in",[507,21985,21986],{"class":517}," backend.target.operation_names\n",[507,21988,21989,21992],{"class":509,"line":559},[507,21990,21991],{"class":517},"):  ",[507,21993,21994],{"class":562},"# Needed as \"op_name\" could be \"if_else\"\n",[507,21996,21997,22000,22003,22006,22009,22011,22014],{"class":509,"line":566},[507,21998,21999],{"class":517},"    backend.target.",[507,22001,22002],{"class":576},"add_instruction",[507,22004,22005],{"class":517},"(IfElseOp, ",[507,22007,22008],{"class":2155},"name",[507,22010,573],{"class":572},[507,22012,22013],{"class":730},"\"if_else\"",[507,22015,587],{"class":517},[507,22017,22018,22020],{"class":509,"line":590},[507,22019,8525],{"class":572},[507,22021,22022],{"class":517},"(backend.name)\n",[2513,22024,22026],{"id":22025},"transpiling-to-a-qpu","Transpiling to a QPU",[18,22028,22029],{},"First, let's pick subsets of the coupling map with \"good\" performing qubits (where \"good\" is pretty arbitrary here, we mostly want to avoid really poor performing qubits) and create a new target for transpilation",[498,22031,22033],{"className":500,"code":22032,"language":502,"meta":104,"style":104},"target = backend.target\ncmap = target.build_coupling_map(filter_idle_qubits=True)\ncmap_list = list(cmap.get_edges())\n\ncust_cmap_list = copy.deepcopy(cmap_list)\nfor q in range(target.num_qubits):\n    meas_err = target[\"measure\"][(q,)].error\n    t2 = target.qubit_properties[q].t2 * 1e6\n    if meas_err > 0.02 or t2 \u003C 100:\n        for q_pair in cmap_list:\n            if q in q_pair:\n                try:\n                    cust_cmap_list.remove(q_pair)\n                except:\n                    continue\n\nfor q in cmap_list:\n    op_name = list(target.operation_names_for_qargs(q))[0]\n    twoq_gate_err = target[f\"{op_name}\"][q].error\n    if twoq_gate_err > 0.005:\n        for q_pair in cmap_list:\n            if q == q_pair:\n                try:\n                    cust_cmap_list.remove(q)\n                except:\n                    continue\n\n\ncust_cmap = CouplingMap(cust_cmap_list)\ncust_target = Target.from_configuration(\n    basis_gates=backend.configuration().basis_gates,\n    coupling_map=cust_cmap,\n)\n",[504,22034,22035,22045,22069,22088,22092,22108,22122,22138,22153,22178,22190,22201,22208,22219,22226,22231,22235,22245,22267,22294,22308,22318,22328,22334,22343,22349,22353,22357,22361,22374,22389,22405,22415],{"__ignoreMap":104},[507,22036,22037,22040,22042],{"class":509,"line":510},[507,22038,22039],{"class":517},"target ",[507,22041,573],{"class":572},[507,22043,22044],{"class":517}," backend.target\n",[507,22046,22047,22050,22052,22055,22058,22060,22063,22065,22067],{"class":509,"line":105},[507,22048,22049],{"class":517},"cmap ",[507,22051,573],{"class":572},[507,22053,22054],{"class":517}," target.",[507,22056,22057],{"class":576},"build_coupling_map",[507,22059,580],{"class":517},[507,22061,22062],{"class":2155},"filter_idle_qubits",[507,22064,573],{"class":572},[507,22066,13878],{"class":583},[507,22068,587],{"class":517},[507,22070,22071,22074,22076,22079,22082,22085],{"class":509,"line":540},[507,22072,22073],{"class":517},"cmap_list ",[507,22075,573],{"class":572},[507,22077,22078],{"class":572}," list",[507,22080,22081],{"class":517},"(cmap.",[507,22083,22084],{"class":576},"get_edges",[507,22086,22087],{"class":517},"())\n",[507,22089,22090],{"class":509,"line":553},[507,22091,556],{"emptyLinePlaceholder":133},[507,22093,22094,22097,22099,22102,22105],{"class":509,"line":559},[507,22095,22096],{"class":517},"cust_cmap_list ",[507,22098,573],{"class":572},[507,22100,22101],{"class":517}," copy.",[507,22103,22104],{"class":576},"deepcopy",[507,22106,22107],{"class":517},"(cmap_list)\n",[507,22109,22110,22112,22115,22117,22119],{"class":509,"line":566},[507,22111,1630],{"class":513},[507,22113,22114],{"class":517}," q ",[507,22116,1636],{"class":513},[507,22118,8221],{"class":572},[507,22120,22121],{"class":517},"(target.num_qubits):\n",[507,22123,22124,22127,22129,22132,22135],{"class":509,"line":590},[507,22125,22126],{"class":517},"    meas_err ",[507,22128,573],{"class":572},[507,22130,22131],{"class":517}," target[",[507,22133,22134],{"class":730},"\"measure\"",[507,22136,22137],{"class":517},"][(q,)].error\n",[507,22139,22140,22143,22145,22148,22150],{"class":509,"line":610},[507,22141,22142],{"class":517},"    t2 ",[507,22144,573],{"class":572},[507,22146,22147],{"class":517}," target.qubit_properties[q].t2 ",[507,22149,2391],{"class":572},[507,22151,22152],{"class":583}," 1e6\n",[507,22154,22155,22157,22160,22162,22165,22168,22171,22173,22176],{"class":509,"line":634},[507,22156,1717],{"class":513},[507,22158,22159],{"class":517}," meas_err ",[507,22161,1651],{"class":572},[507,22163,22164],{"class":583}," 0.02",[507,22166,22167],{"class":513}," or",[507,22169,22170],{"class":517}," t2 ",[507,22172,5677],{"class":572},[507,22174,22175],{"class":583}," 100",[507,22177,1728],{"class":517},[507,22179,22180,22182,22185,22187],{"class":509,"line":661},[507,22181,2267],{"class":513},[507,22183,22184],{"class":517}," q_pair ",[507,22186,1636],{"class":513},[507,22188,22189],{"class":517}," cmap_list:\n",[507,22191,22192,22194,22196,22198],{"class":509,"line":678},[507,22193,2298],{"class":513},[507,22195,22114],{"class":517},[507,22197,1636],{"class":513},[507,22199,22200],{"class":517}," q_pair:\n",[507,22202,22203,22206],{"class":509,"line":683},[507,22204,22205],{"class":513},"                try",[507,22207,1728],{"class":517},[507,22209,22210,22213,22216],{"class":509,"line":697},[507,22211,22212],{"class":517},"                    cust_cmap_list.",[507,22214,22215],{"class":576},"remove",[507,22217,22218],{"class":517},"(q_pair)\n",[507,22220,22221,22224],{"class":509,"line":710},[507,22222,22223],{"class":513},"                except",[507,22225,1728],{"class":517},[507,22227,22228],{"class":509,"line":715},[507,22229,22230],{"class":513},"                    continue\n",[507,22232,22233],{"class":509,"line":721},[507,22234,556],{"emptyLinePlaceholder":133},[507,22236,22237,22239,22241,22243],{"class":509,"line":736},[507,22238,1630],{"class":513},[507,22240,22114],{"class":517},[507,22242,1636],{"class":513},[507,22244,22189],{"class":517},[507,22246,22247,22250,22252,22254,22257,22260,22263,22265],{"class":509,"line":748},[507,22248,22249],{"class":517},"    op_name ",[507,22251,573],{"class":572},[507,22253,22078],{"class":572},[507,22255,22256],{"class":517},"(target.",[507,22258,22259],{"class":576},"operation_names_for_qargs",[507,22261,22262],{"class":517},"(q))[",[507,22264,601],{"class":583},[507,22266,1794],{"class":517},[507,22268,22269,22272,22274,22276,22279,22282,22284,22287,22289,22291],{"class":509,"line":761},[507,22270,22271],{"class":517},"    twoq_gate_err ",[507,22273,573],{"class":572},[507,22275,22131],{"class":517},[507,22277,22278],{"class":513},"f",[507,22280,22281],{"class":730},"\"",[507,22283,2810],{"class":583},[507,22285,22286],{"class":517},"op_name",[507,22288,2872],{"class":583},[507,22290,22281],{"class":730},[507,22292,22293],{"class":517},"][q].error\n",[507,22295,22296,22298,22301,22303,22306],{"class":509,"line":775},[507,22297,1717],{"class":513},[507,22299,22300],{"class":517}," twoq_gate_err ",[507,22302,1651],{"class":572},[507,22304,22305],{"class":583}," 0.005",[507,22307,1728],{"class":517},[507,22309,22310,22312,22314,22316],{"class":509,"line":784},[507,22311,2267],{"class":513},[507,22313,22184],{"class":517},[507,22315,1636],{"class":513},[507,22317,22189],{"class":517},[507,22319,22320,22322,22324,22326],{"class":509,"line":796},[507,22321,2298],{"class":513},[507,22323,22114],{"class":517},[507,22325,1723],{"class":572},[507,22327,22200],{"class":517},[507,22329,22330,22332],{"class":509,"line":809},[507,22331,22205],{"class":513},[507,22333,1728],{"class":517},[507,22335,22336,22338,22340],{"class":509,"line":1352},[507,22337,22212],{"class":517},[507,22339,22215],{"class":576},[507,22341,22342],{"class":517},"(q)\n",[507,22344,22345,22347],{"class":509,"line":1357},[507,22346,22223],{"class":513},[507,22348,1728],{"class":517},[507,22350,22351],{"class":509,"line":1362},[507,22352,22230],{"class":513},[507,22354,22355],{"class":509,"line":1367},[507,22356,556],{"emptyLinePlaceholder":133},[507,22358,22359],{"class":509,"line":1379},[507,22360,556],{"emptyLinePlaceholder":133},[507,22362,22363,22366,22368,22371],{"class":509,"line":1389},[507,22364,22365],{"class":517},"cust_cmap ",[507,22367,573],{"class":572},[507,22369,22370],{"class":576}," CouplingMap",[507,22372,22373],{"class":517},"(cust_cmap_list)\n",[507,22375,22376,22379,22381,22384,22387],{"class":509,"line":1397},[507,22377,22378],{"class":517},"cust_target ",[507,22380,573],{"class":572},[507,22382,22383],{"class":517}," Target.",[507,22385,22386],{"class":576},"from_configuration",[507,22388,1376],{"class":517},[507,22390,22391,22394,22396,22399,22402],{"class":509,"line":1412},[507,22392,22393],{"class":2155},"    basis_gates",[507,22395,573],{"class":572},[507,22397,22398],{"class":517},"backend.",[507,22400,22401],{"class":576},"configuration",[507,22403,22404],{"class":517},"().basis_gates,\n",[507,22406,22407,22410,22412],{"class":509,"line":1431},[507,22408,22409],{"class":2155},"    coupling_map",[507,22411,573],{"class":572},[507,22413,22414],{"class":517},"cust_cmap,\n",[507,22416,22417],{"class":509,"line":1449},[507,22418,587],{"class":517},[18,22420,22421],{},"Then transpile the virtual circuit to the best physical layout in this new target",[498,22423,22425],{"className":500,"code":22424,"language":502,"meta":104,"style":104},"basis_gates = list(target.operation_names)\npm = generate_preset_pass_manager(\n    optimization_level=3,\n    target=cust_target,\n    basis_gates=basis_gates,\n)\n\nqc_trans = pm.run(qc)\n\nprint(\"depth\", qc_trans.depth(lambda x: x[0].num_qubits == 2))\nprint(\"num 2q ops\", qc_trans.count_ops())\nprint(\n    \"physical qubits\",\n    sorted(\n        [\n            idx\n            for idx, qb in qc_trans.layout.initial_layout.get_physical_bits().items()\n            if qb._register.name != \"ancilla\"\n        ]\n    ),\n)\n",[504,22426,22427,22439,22451,22462,22472,22481,22485,22489,22504,22508,22541,22557,22563,22570,22577,22582,22587,22610,22623,22628,22633],{"__ignoreMap":104},[507,22428,22429,22432,22434,22436],{"class":509,"line":510},[507,22430,22431],{"class":517},"basis_gates ",[507,22433,573],{"class":572},[507,22435,22078],{"class":572},[507,22437,22438],{"class":517},"(target.operation_names)\n",[507,22440,22441,22444,22446,22449],{"class":509,"line":105},[507,22442,22443],{"class":517},"pm ",[507,22445,573],{"class":572},[507,22447,22448],{"class":576}," generate_preset_pass_manager",[507,22450,1376],{"class":517},[507,22452,22453,22456,22458,22460],{"class":509,"line":540},[507,22454,22455],{"class":2155},"    optimization_level",[507,22457,573],{"class":572},[507,22459,8226],{"class":583},[507,22461,1409],{"class":517},[507,22463,22464,22467,22469],{"class":509,"line":553},[507,22465,22466],{"class":2155},"    target",[507,22468,573],{"class":572},[507,22470,22471],{"class":517},"cust_target,\n",[507,22473,22474,22476,22478],{"class":509,"line":559},[507,22475,22393],{"class":2155},[507,22477,573],{"class":572},[507,22479,22480],{"class":517},"basis_gates,\n",[507,22482,22483],{"class":509,"line":566},[507,22484,587],{"class":517},[507,22486,22487],{"class":509,"line":590},[507,22488,556],{"emptyLinePlaceholder":133},[507,22490,22491,22494,22496,22499,22502],{"class":509,"line":610},[507,22492,22493],{"class":517},"qc_trans ",[507,22495,573],{"class":572},[507,22497,22498],{"class":517}," pm.",[507,22500,22501],{"class":576},"run",[507,22503,694],{"class":517},[507,22505,22506],{"class":509,"line":634},[507,22507,556],{"emptyLinePlaceholder":133},[507,22509,22510,22512,22514,22517,22520,22522,22524,22526,22528,22530,22532,22534,22536,22538],{"class":509,"line":661},[507,22511,8525],{"class":572},[507,22513,580],{"class":517},[507,22515,22516],{"class":730},"\"depth\"",[507,22518,22519],{"class":517},", qc_trans.",[507,22521,14179],{"class":576},[507,22523,580],{"class":517},[507,22525,14184],{"class":513},[507,22527,14187],{"class":1382},[507,22529,14190],{"class":517},[507,22531,601],{"class":583},[507,22533,14195],{"class":517},[507,22535,1723],{"class":572},[507,22537,2316],{"class":583},[507,22539,22540],{"class":517},"))\n",[507,22542,22543,22545,22547,22550,22552,22555],{"class":509,"line":678},[507,22544,8525],{"class":572},[507,22546,580],{"class":517},[507,22548,22549],{"class":730},"\"num 2q ops\"",[507,22551,22519],{"class":517},[507,22553,22554],{"class":576},"count_ops",[507,22556,22087],{"class":517},[507,22558,22559,22561],{"class":509,"line":683},[507,22560,8525],{"class":572},[507,22562,1376],{"class":517},[507,22564,22565,22568],{"class":509,"line":697},[507,22566,22567],{"class":730},"    \"physical qubits\"",[507,22569,1409],{"class":517},[507,22571,22572,22575],{"class":509,"line":710},[507,22573,22574],{"class":572},"    sorted",[507,22576,1376],{"class":517},[507,22578,22579],{"class":509,"line":715},[507,22580,22581],{"class":517},"        [\n",[507,22583,22584],{"class":509,"line":721},[507,22585,22586],{"class":517},"            idx\n",[507,22588,22589,22592,22595,22597,22600,22603,22605,22608],{"class":509,"line":736},[507,22590,22591],{"class":513},"            for",[507,22593,22594],{"class":517}," idx, qb ",[507,22596,1636],{"class":513},[507,22598,22599],{"class":517}," qc_trans.layout.initial_layout.",[507,22601,22602],{"class":576},"get_physical_bits",[507,22604,13983],{"class":517},[507,22606,22607],{"class":576},"items",[507,22609,781],{"class":517},[507,22611,22612,22614,22617,22620],{"class":509,"line":748},[507,22613,2298],{"class":513},[507,22615,22616],{"class":517}," qb._register.name ",[507,22618,22619],{"class":572},"!=",[507,22621,22622],{"class":730}," \"ancilla\"\n",[507,22624,22625],{"class":509,"line":761},[507,22626,22627],{"class":517},"        ]\n",[507,22629,22630],{"class":509,"line":775},[507,22631,22632],{"class":517},"    ),\n",[507,22634,22635],{"class":509,"line":784},[507,22636,587],{"class":517},[498,22638,22641],{"className":22639,"code":22640,"language":7039,"meta":104},[8531],"depth 52\nnum 2q ops OrderedDict([('rz', 2058), ('sx', 1703), ('cz', 728), ('x', 84), ('barrier', 8)])\nphysical qubits [91, 92, 93, 94, 95, 98, 99, 108, 109, 110, 111, 113, 114, 115, 119, 127, 132, 133, 134, 135, 137, 139, 147, 148, 149, 150, 151, 152, 153, 154, 155]\n",[504,22642,22640],{"__ignoreMap":104},[2513,22644,22646],{"id":22645},"create-pubs-for-execution-with-estimator","Create PUBs for execution with Estimator",[498,22648,22650],{"className":500,"code":22649,"language":502,"meta":104,"style":104},"# Define observables to measure for S\nobservable_S_real = \"I\" * (n_qubits) + \"X\"\nobservable_S_imag = \"I\" * (n_qubits) + \"Y\"\n\nobservable_op_real = SparsePauliOp(\n    observable_S_real\n)  # define a sparse pauli operator for the observable\nobservable_op_imag = SparsePauliOp(observable_S_imag)\n\nlayout = qc_trans.layout  # get layout of transpiled circuit\nobservable_op_real = observable_op_real.apply_layout(\n    layout\n)  # apply physical layout to the observable\nobservable_op_imag = observable_op_imag.apply_layout(layout)\nobservable_S_real = (\n    observable_op_real.paulis.to_labels()\n)  # get the label of the physical observable\nobservable_S_imag = observable_op_imag.paulis.to_labels()\n\nobservables_S = [[observable_S_real], [observable_S_imag]]\n\n\n# Define observables to measure for H\n# Hamiltonian terms to measure\nobservable_list = []\nfor pauli, coeff in zip(H_op.paulis, H_op.coeffs):\n    # print(pauli)\n    observable_H_real = pauli[::-1].to_label() + \"X\"\n    observable_H_imag = pauli[::-1].to_label() + \"Y\"\n    observable_list.append([observable_H_real])\n    observable_list.append([observable_H_imag])\n\nlayout = qc_trans.layout\n\nobservable_trans_list = []\nfor observable in observable_list:\n    observable_op = SparsePauliOp(observable)\n    observable_op = observable_op.apply_layout(layout)\n    observable_trans_list.append([observable_op.paulis.to_labels()])\n\nobservables_H = observable_trans_list\n\n\n# Define a sweep over parameter values\nparams = np.vstack(parameters).T\n\n\n# Estimate the expectation value for all combinations of\n# observables and parameter values, where the pub result will have\n# shape (# observables, # parameter values).\npub = (qc_trans, observables_S + observables_H, params)\n",[504,22651,22652,22657,22676,22693,22697,22709,22714,22722,22734,22738,22751,22765,22770,22777,22791,22799,22809,22816,22829,22833,22843,22847,22851,22856,22861,22870,22884,22889,22915,22938,22948,22957,22961,22970,22974,22983,22995,23007,23020,23035,23039,23049,23053,23057,23062,23077,23081,23085,23090,23095,23100],{"__ignoreMap":104},[507,22653,22654],{"class":509,"line":510},[507,22655,22656],{"class":562},"# Define observables to measure for S\n",[507,22658,22659,22662,22664,22667,22669,22672,22674],{"class":509,"line":105},[507,22660,22661],{"class":517},"observable_S_real ",[507,22663,573],{"class":572},[507,22665,22666],{"class":730}," \"I\"",[507,22668,8229],{"class":572},[507,22670,22671],{"class":517}," (n_qubits) ",[507,22673,2107],{"class":572},[507,22675,8321],{"class":730},[507,22677,22678,22681,22683,22685,22687,22689,22691],{"class":509,"line":540},[507,22679,22680],{"class":517},"observable_S_imag ",[507,22682,573],{"class":572},[507,22684,22666],{"class":730},[507,22686,8229],{"class":572},[507,22688,22671],{"class":517},[507,22690,2107],{"class":572},[507,22692,8388],{"class":730},[507,22694,22695],{"class":509,"line":553},[507,22696,556],{"emptyLinePlaceholder":133},[507,22698,22699,22702,22704,22707],{"class":509,"line":559},[507,22700,22701],{"class":517},"observable_op_real ",[507,22703,573],{"class":572},[507,22705,22706],{"class":576}," SparsePauliOp",[507,22708,1376],{"class":517},[507,22710,22711],{"class":509,"line":566},[507,22712,22713],{"class":517},"    observable_S_real\n",[507,22715,22716,22719],{"class":509,"line":590},[507,22717,22718],{"class":517},")  ",[507,22720,22721],{"class":562},"# define a sparse pauli operator for the observable\n",[507,22723,22724,22727,22729,22731],{"class":509,"line":610},[507,22725,22726],{"class":517},"observable_op_imag ",[507,22728,573],{"class":572},[507,22730,22706],{"class":576},[507,22732,22733],{"class":517},"(observable_S_imag)\n",[507,22735,22736],{"class":509,"line":634},[507,22737,556],{"emptyLinePlaceholder":133},[507,22739,22740,22743,22745,22748],{"class":509,"line":661},[507,22741,22742],{"class":517},"layout ",[507,22744,573],{"class":572},[507,22746,22747],{"class":517}," qc_trans.layout  ",[507,22749,22750],{"class":562},"# get layout of transpiled circuit\n",[507,22752,22753,22755,22757,22760,22763],{"class":509,"line":678},[507,22754,22701],{"class":517},[507,22756,573],{"class":572},[507,22758,22759],{"class":517}," observable_op_real.",[507,22761,22762],{"class":576},"apply_layout",[507,22764,1376],{"class":517},[507,22766,22767],{"class":509,"line":683},[507,22768,22769],{"class":517},"    layout\n",[507,22771,22772,22774],{"class":509,"line":697},[507,22773,22718],{"class":517},[507,22775,22776],{"class":562},"# apply physical layout to the observable\n",[507,22778,22779,22781,22783,22786,22788],{"class":509,"line":710},[507,22780,22726],{"class":517},[507,22782,573],{"class":572},[507,22784,22785],{"class":517}," observable_op_imag.",[507,22787,22762],{"class":576},[507,22789,22790],{"class":517},"(layout)\n",[507,22792,22793,22795,22797],{"class":509,"line":715},[507,22794,22661],{"class":517},[507,22796,573],{"class":572},[507,22798,1334],{"class":517},[507,22800,22801,22804,22807],{"class":509,"line":721},[507,22802,22803],{"class":517},"    observable_op_real.paulis.",[507,22805,22806],{"class":576},"to_labels",[507,22808,781],{"class":517},[507,22810,22811,22813],{"class":509,"line":736},[507,22812,22718],{"class":517},[507,22814,22815],{"class":562},"# get the label of the physical observable\n",[507,22817,22818,22820,22822,22825,22827],{"class":509,"line":748},[507,22819,22680],{"class":517},[507,22821,573],{"class":572},[507,22823,22824],{"class":517}," observable_op_imag.paulis.",[507,22826,22806],{"class":576},[507,22828,781],{"class":517},[507,22830,22831],{"class":509,"line":761},[507,22832,556],{"emptyLinePlaceholder":133},[507,22834,22835,22838,22840],{"class":509,"line":775},[507,22836,22837],{"class":517},"observables_S ",[507,22839,573],{"class":572},[507,22841,22842],{"class":517}," [[observable_S_real], [observable_S_imag]]\n",[507,22844,22845],{"class":509,"line":784},[507,22846,556],{"emptyLinePlaceholder":133},[507,22848,22849],{"class":509,"line":796},[507,22850,556],{"emptyLinePlaceholder":133},[507,22852,22853],{"class":509,"line":809},[507,22854,22855],{"class":562},"# Define observables to measure for H\n",[507,22857,22858],{"class":509,"line":1352},[507,22859,22860],{"class":562},"# Hamiltonian terms to measure\n",[507,22862,22863,22866,22868],{"class":509,"line":1357},[507,22864,22865],{"class":517},"observable_list ",[507,22867,573],{"class":572},[507,22869,1910],{"class":517},[507,22871,22872,22874,22877,22879,22881],{"class":509,"line":1362},[507,22873,1630],{"class":513},[507,22875,22876],{"class":517}," pauli, coeff ",[507,22878,1636],{"class":513},[507,22880,1639],{"class":572},[507,22882,22883],{"class":517},"(H_op.paulis, H_op.coeffs):\n",[507,22885,22886],{"class":509,"line":1367},[507,22887,22888],{"class":562},"    # print(pauli)\n",[507,22890,22891,22894,22896,22899,22901,22903,22906,22909,22911,22913],{"class":509,"line":1379},[507,22892,22893],{"class":517},"    observable_H_real ",[507,22895,573],{"class":572},[507,22897,22898],{"class":517}," pauli[::",[507,22900,2367],{"class":572},[507,22902,625],{"class":583},[507,22904,22905],{"class":517},"].",[507,22907,22908],{"class":576},"to_label",[507,22910,1677],{"class":517},[507,22912,2107],{"class":572},[507,22914,8321],{"class":730},[507,22916,22917,22920,22922,22924,22926,22928,22930,22932,22934,22936],{"class":509,"line":1389},[507,22918,22919],{"class":517},"    observable_H_imag ",[507,22921,573],{"class":572},[507,22923,22898],{"class":517},[507,22925,2367],{"class":572},[507,22927,625],{"class":583},[507,22929,22905],{"class":517},[507,22931,22908],{"class":576},[507,22933,1677],{"class":517},[507,22935,2107],{"class":572},[507,22937,8388],{"class":730},[507,22939,22940,22943,22945],{"class":509,"line":1397},[507,22941,22942],{"class":517},"    observable_list.",[507,22944,1939],{"class":576},[507,22946,22947],{"class":517},"([observable_H_real])\n",[507,22949,22950,22952,22954],{"class":509,"line":1412},[507,22951,22942],{"class":517},[507,22953,1939],{"class":576},[507,22955,22956],{"class":517},"([observable_H_imag])\n",[507,22958,22959],{"class":509,"line":1431},[507,22960,556],{"emptyLinePlaceholder":133},[507,22962,22963,22965,22967],{"class":509,"line":1449},[507,22964,22742],{"class":517},[507,22966,573],{"class":572},[507,22968,22969],{"class":517}," qc_trans.layout\n",[507,22971,22972],{"class":509,"line":1465},[507,22973,556],{"emptyLinePlaceholder":133},[507,22975,22976,22979,22981],{"class":509,"line":1471},[507,22977,22978],{"class":517},"observable_trans_list ",[507,22980,573],{"class":572},[507,22982,1910],{"class":517},[507,22984,22985,22987,22990,22992],{"class":509,"line":1477},[507,22986,1630],{"class":513},[507,22988,22989],{"class":517}," observable ",[507,22991,1636],{"class":513},[507,22993,22994],{"class":517}," observable_list:\n",[507,22996,22997,23000,23002,23004],{"class":509,"line":1482},[507,22998,22999],{"class":517},"    observable_op ",[507,23001,573],{"class":572},[507,23003,22706],{"class":576},[507,23005,23006],{"class":517},"(observable)\n",[507,23008,23009,23011,23013,23016,23018],{"class":509,"line":1488},[507,23010,22999],{"class":517},[507,23012,573],{"class":572},[507,23014,23015],{"class":517}," observable_op.",[507,23017,22762],{"class":576},[507,23019,22790],{"class":517},[507,23021,23022,23025,23027,23030,23032],{"class":509,"line":1494},[507,23023,23024],{"class":517},"    observable_trans_list.",[507,23026,1939],{"class":576},[507,23028,23029],{"class":517},"([observable_op.paulis.",[507,23031,22806],{"class":576},[507,23033,23034],{"class":517},"()])\n",[507,23036,23037],{"class":509,"line":1500},[507,23038,556],{"emptyLinePlaceholder":133},[507,23040,23041,23044,23046],{"class":509,"line":1506},[507,23042,23043],{"class":517},"observables_H ",[507,23045,573],{"class":572},[507,23047,23048],{"class":517}," observable_trans_list\n",[507,23050,23051],{"class":509,"line":1512},[507,23052,556],{"emptyLinePlaceholder":133},[507,23054,23055],{"class":509,"line":1518},[507,23056,556],{"emptyLinePlaceholder":133},[507,23058,23059],{"class":509,"line":1524},[507,23060,23061],{"class":562},"# Define a sweep over parameter values\n",[507,23063,23064,23067,23069,23071,23074],{"class":509,"line":1530},[507,23065,23066],{"class":517},"params ",[507,23068,573],{"class":572},[507,23070,1616],{"class":517},[507,23072,23073],{"class":576},"vstack",[507,23075,23076],{"class":517},"(parameters).T\n",[507,23078,23079],{"class":509,"line":1536},[507,23080,556],{"emptyLinePlaceholder":133},[507,23082,23083],{"class":509,"line":1542},[507,23084,556],{"emptyLinePlaceholder":133},[507,23086,23087],{"class":509,"line":1548},[507,23088,23089],{"class":562},"# Estimate the expectation value for all combinations of\n",[507,23091,23092],{"class":509,"line":1553},[507,23093,23094],{"class":562},"# observables and parameter values, where the pub result will have\n",[507,23096,23097],{"class":509,"line":1559},[507,23098,23099],{"class":562},"# shape (# observables, # parameter values).\n",[507,23101,23102,23105,23107,23110,23112],{"class":509,"line":1565},[507,23103,23104],{"class":517},"pub ",[507,23106,573],{"class":572},[507,23108,23109],{"class":517}," (qc_trans, observables_S ",[507,23111,2107],{"class":572},[507,23113,23114],{"class":517}," observables_H, params)\n",[2513,23116,23118],{"id":23117},"run-circuits","Run circuits",[18,23120,23121,23122,23173],{},"Circuits for ",[507,23123,23125,23143],{"className":23124},[2523],[507,23126,23128],{"className":23127},[2527],[2529,23129,23130],{"xmlns":2531},[2533,23131,23132,23140],{},[2536,23133,23134,23136,23138],{},[2542,23135,3298],{},[2689,23137,573],{},[2693,23139,601],{},[2549,23141,23142],{"encoding":2551},"t=0",[507,23144,23146,23164],{"className":23145,"ariaHidden":2557},[2556],[507,23147,23149,23152,23155,23158,23161],{"className":23148},[2561],[507,23150],{"className":23151,"style":20623},[2565],[507,23153,3298],{"className":23154},[2570,2611],[507,23156],{"className":23157,"style":2919},[2714],[507,23159,573],{"className":23160},[2923],[507,23162],{"className":23163,"style":2919},[2714],[507,23165,23167,23170],{"className":23166},[2561],[507,23168],{"className":23169,"style":2729},[2565],[507,23171,601],{"className":23172},[2570]," are classically calculable",[498,23175,23177],{"className":500,"code":23176,"language":502,"meta":104,"style":104},"qc_cliff = qc.assign_parameters({t: 0})\n\n\n# Get expectation values from experiment\nS_expval_real = StabilizerState(qc_cliff).expectation_value(\n    Pauli(\"I\" * (n_qubits) + \"X\")\n)\nS_expval_imag = StabilizerState(qc_cliff).expectation_value(\n    Pauli(\"I\" * (n_qubits) + \"Y\")\n)\n\n# Get expectation values\nS_expval = S_expval_real + 1j * S_expval_imag\n\nH_expval = 0\nfor obs_idx, (pauli, coeff) in enumerate(zip(H_op.paulis, H_op.coeffs)):\n    # Get expectation values from experiment\n    expval_real = StabilizerState(qc_cliff).expectation_value(\n        Pauli(pauli[::-1].to_label() + \"X\")\n    )\n    expval_imag = StabilizerState(qc_cliff).expectation_value(\n        Pauli(pauli[::-1].to_label() + \"Y\")\n    )\n    expval = expval_real + 1j * expval_imag\n\n    # Fill-in matrix elements\n    H_expval += coeff * expval\n\n\nprint(H_expval)\n",[504,23178,23179,23200,23204,23208,23213,23231,23251,23255,23270,23289,23293,23297,23302,23323,23327,23336,23355,23360,23375,23399,23403,23418,23440,23444,23465,23469,23474,23489,23493,23497],{"__ignoreMap":104},[507,23180,23181,23184,23186,23189,23192,23195,23197],{"class":509,"line":510},[507,23182,23183],{"class":517},"qc_cliff ",[507,23185,573],{"class":572},[507,23187,23188],{"class":517}," qc.",[507,23190,23191],{"class":576},"assign_parameters",[507,23193,23194],{"class":517},"({t: ",[507,23196,601],{"class":583},[507,23198,23199],{"class":517},"})\n",[507,23201,23202],{"class":509,"line":105},[507,23203,556],{"emptyLinePlaceholder":133},[507,23205,23206],{"class":509,"line":540},[507,23207,556],{"emptyLinePlaceholder":133},[507,23209,23210],{"class":509,"line":553},[507,23211,23212],{"class":562},"# Get expectation values from experiment\n",[507,23214,23215,23218,23220,23223,23226,23229],{"class":509,"line":559},[507,23216,23217],{"class":517},"S_expval_real ",[507,23219,573],{"class":572},[507,23221,23222],{"class":576}," StabilizerState",[507,23224,23225],{"class":517},"(qc_cliff).",[507,23227,23228],{"class":576},"expectation_value",[507,23230,1376],{"class":517},[507,23232,23233,23236,23238,23240,23242,23244,23246,23249],{"class":509,"line":566},[507,23234,23235],{"class":576},"    Pauli",[507,23237,580],{"class":517},[507,23239,8203],{"class":730},[507,23241,8229],{"class":572},[507,23243,22671],{"class":517},[507,23245,2107],{"class":572},[507,23247,23248],{"class":730}," \"X\"",[507,23250,587],{"class":517},[507,23252,23253],{"class":509,"line":590},[507,23254,587],{"class":517},[507,23256,23257,23260,23262,23264,23266,23268],{"class":509,"line":610},[507,23258,23259],{"class":517},"S_expval_imag ",[507,23261,573],{"class":572},[507,23263,23222],{"class":576},[507,23265,23225],{"class":517},[507,23267,23228],{"class":576},[507,23269,1376],{"class":517},[507,23271,23272,23274,23276,23278,23280,23282,23284,23287],{"class":509,"line":634},[507,23273,23235],{"class":576},[507,23275,580],{"class":517},[507,23277,8203],{"class":730},[507,23279,8229],{"class":572},[507,23281,22671],{"class":517},[507,23283,2107],{"class":572},[507,23285,23286],{"class":730}," \"Y\"",[507,23288,587],{"class":517},[507,23290,23291],{"class":509,"line":661},[507,23292,587],{"class":517},[507,23294,23295],{"class":509,"line":678},[507,23296,556],{"emptyLinePlaceholder":133},[507,23298,23299],{"class":509,"line":683},[507,23300,23301],{"class":562},"# Get expectation values\n",[507,23303,23304,23307,23309,23312,23314,23316,23318,23320],{"class":509,"line":697},[507,23305,23306],{"class":517},"S_expval ",[507,23308,573],{"class":572},[507,23310,23311],{"class":517}," S_expval_real ",[507,23313,2107],{"class":572},[507,23315,1426],{"class":583},[507,23317,2372],{"class":513},[507,23319,8229],{"class":572},[507,23321,23322],{"class":517}," S_expval_imag\n",[507,23324,23325],{"class":509,"line":710},[507,23326,556],{"emptyLinePlaceholder":133},[507,23328,23329,23332,23334],{"class":509,"line":715},[507,23330,23331],{"class":517},"H_expval ",[507,23333,573],{"class":572},[507,23335,2246],{"class":583},[507,23337,23338,23340,23343,23345,23347,23349,23352],{"class":509,"line":721},[507,23339,1630],{"class":513},[507,23341,23342],{"class":517}," obs_idx, (pauli, coeff) ",[507,23344,1636],{"class":513},[507,23346,1957],{"class":572},[507,23348,580],{"class":517},[507,23350,23351],{"class":572},"zip",[507,23353,23354],{"class":517},"(H_op.paulis, H_op.coeffs)):\n",[507,23356,23357],{"class":509,"line":736},[507,23358,23359],{"class":562},"    # Get expectation values from experiment\n",[507,23361,23362,23365,23367,23369,23371,23373],{"class":509,"line":748},[507,23363,23364],{"class":517},"    expval_real ",[507,23366,573],{"class":572},[507,23368,23222],{"class":576},[507,23370,23225],{"class":517},[507,23372,23228],{"class":576},[507,23374,1376],{"class":517},[507,23376,23377,23380,23383,23385,23387,23389,23391,23393,23395,23397],{"class":509,"line":761},[507,23378,23379],{"class":576},"        Pauli",[507,23381,23382],{"class":517},"(pauli[::",[507,23384,2367],{"class":572},[507,23386,625],{"class":583},[507,23388,22905],{"class":517},[507,23390,22908],{"class":576},[507,23392,1677],{"class":517},[507,23394,2107],{"class":572},[507,23396,23248],{"class":730},[507,23398,587],{"class":517},[507,23400,23401],{"class":509,"line":775},[507,23402,1660],{"class":517},[507,23404,23405,23408,23410,23412,23414,23416],{"class":509,"line":784},[507,23406,23407],{"class":517},"    expval_imag ",[507,23409,573],{"class":572},[507,23411,23222],{"class":576},[507,23413,23225],{"class":517},[507,23415,23228],{"class":576},[507,23417,1376],{"class":517},[507,23419,23420,23422,23424,23426,23428,23430,23432,23434,23436,23438],{"class":509,"line":796},[507,23421,23379],{"class":576},[507,23423,23382],{"class":517},[507,23425,2367],{"class":572},[507,23427,625],{"class":583},[507,23429,22905],{"class":517},[507,23431,22908],{"class":576},[507,23433,1677],{"class":517},[507,23435,2107],{"class":572},[507,23437,23286],{"class":730},[507,23439,587],{"class":517},[507,23441,23442],{"class":509,"line":809},[507,23443,1660],{"class":517},[507,23445,23446,23449,23451,23454,23456,23458,23460,23462],{"class":509,"line":1352},[507,23447,23448],{"class":517},"    expval ",[507,23450,573],{"class":572},[507,23452,23453],{"class":517}," expval_real ",[507,23455,2107],{"class":572},[507,23457,1426],{"class":583},[507,23459,2372],{"class":513},[507,23461,8229],{"class":572},[507,23463,23464],{"class":517}," expval_imag\n",[507,23466,23467],{"class":509,"line":1357},[507,23468,556],{"emptyLinePlaceholder":133},[507,23470,23471],{"class":509,"line":1362},[507,23472,23473],{"class":562},"    # Fill-in matrix elements\n",[507,23475,23476,23479,23481,23484,23486],{"class":509,"line":1367},[507,23477,23478],{"class":517},"    H_expval ",[507,23480,2285],{"class":572},[507,23482,23483],{"class":517}," coeff ",[507,23485,2391],{"class":572},[507,23487,23488],{"class":517}," expval\n",[507,23490,23491],{"class":509,"line":1379},[507,23492,556],{"emptyLinePlaceholder":133},[507,23494,23495],{"class":509,"line":1389},[507,23496,556],{"emptyLinePlaceholder":133},[507,23498,23499,23501],{"class":509,"line":1397},[507,23500,8525],{"class":572},[507,23502,23503],{"class":517},"(H_expval)\n",[498,23505,23508],{"className":23506,"code":23507,"language":7039,"meta":104},[8531],"(25+0j)\n",[504,23509,23507],{"__ignoreMap":104},[18,23511,23512,23513,10799,23541,23601],{},"Execute circuits for ",[507,23514,23516,23529],{"className":23515},[2523],[507,23517,23519],{"className":23518},[2527],[2529,23520,23521],{"xmlns":2531},[2533,23522,23523,23527],{},[2536,23524,23525],{},[2542,23526,5857],{},[2549,23528,5857],{"encoding":2551},[507,23530,23532],{"className":23531,"ariaHidden":2557},[2556],[507,23533,23535,23538],{"className":23534},[2561],[507,23536],{"className":23537,"style":2566},[2565],[507,23539,5857],{"className":23540,"style":5892},[2570,2611],[507,23542,23544,23561],{"className":23543},[2523],[507,23545,23547],{"className":23546},[2527],[2529,23548,23549],{"xmlns":2531},[2533,23550,23551,23559],{},[2536,23552,23553],{},[4271,23554,23555,23557],{"accent":2557},[2542,23556,3138],{},[2689,23558,4277],{},[2549,23560,4280],{"encoding":2551},[507,23562,23564],{"className":23563,"ariaHidden":2557},[2556],[507,23565,23567,23570],{"className":23566},[2561],[507,23568],{"className":23569,"style":4290},[2565],[507,23571,23573],{"className":23572},[2570,4294],[507,23574,23576],{"className":23575},[2583],[507,23577,23579],{"className":23578},[2587],[507,23580,23582,23590],{"className":23581,"style":4290},[2591],[507,23583,23584,23587],{"style":4306},[507,23585],{"className":23586,"style":4310},[2599],[507,23588,3138],{"className":23589,"style":3153},[2570,2611],[507,23591,23592,23595],{"style":4316},[507,23593],{"className":23594,"style":4310},[2599],[507,23596,23598],{"className":23597,"style":4324},[4323],[507,23599,4277],{"className":23600},[2570]," with Estimator",[498,23603,23605],{"className":500,"code":23604,"language":502,"meta":104,"style":104},"# Experiment options\nnum_randomizations = 300\nnum_randomizations_learning = 30\nshots_per_randomization = 100\nnoise_factors = [1, 1.2, 1.4]\nlearning_pair_depths = [0, 4, 24, 48]\n\n\nexperimental_opts = {}\nexperimental_opts[\"resilience\"] = {\n    \"measure_mitigation\": True,\n    \"measure_noise_learning\": {\n        \"num_randomizations\": num_randomizations_learning,\n        \"shots_per_randomization\": shots_per_randomization,\n    },\n    \"zne_mitigation\": True,\n    \"zne\": {\"noise_factors\": noise_factors},\n    \"layer_noise_learning\": {\n        \"max_layers_to_learn\": 10,\n        \"layer_pair_depths\": learning_pair_depths,\n        \"shots_per_randomization\": shots_per_randomization,\n        \"num_randomizations\": num_randomizations_learning,\n    },\n    \"zne\": {\n        \"amplifier\": \"pea\",\n        \"extrapolated_noise_factors\": [0] + noise_factors,\n    },\n}\nexperimental_opts[\"twirling\"] = {\n    \"num_randomizations\": num_randomizations,\n    \"shots_per_randomization\": shots_per_randomization,\n    \"strategy\": \"all\",\n}\n\nestimator = Estimator(mode=backend, options=experimental_opts)\n\n\njob = estimator.run([pub])\n",[504,23606,23607,23612,23622,23631,23641,23664,23691,23695,23699,23709,23724,23735,23743,23751,23759,23764,23775,23789,23796,23808,23816,23822,23828,23832,23838,23850,23867,23871,23876,23889,23897,23904,23916,23920,23924,23952,23956,23960],{"__ignoreMap":104},[507,23608,23609],{"class":509,"line":510},[507,23610,23611],{"class":562},"# Experiment options\n",[507,23613,23614,23617,23619],{"class":509,"line":105},[507,23615,23616],{"class":517},"num_randomizations ",[507,23618,573],{"class":572},[507,23620,23621],{"class":583}," 300\n",[507,23623,23624,23627,23629],{"class":509,"line":540},[507,23625,23626],{"class":517},"num_randomizations_learning ",[507,23628,573],{"class":572},[507,23630,8169],{"class":583},[507,23632,23633,23636,23638],{"class":509,"line":553},[507,23634,23635],{"class":517},"shots_per_randomization ",[507,23637,573],{"class":572},[507,23639,23640],{"class":583}," 100\n",[507,23642,23643,23646,23648,23650,23652,23654,23657,23659,23662],{"class":509,"line":559},[507,23644,23645],{"class":517},"noise_factors ",[507,23647,573],{"class":572},[507,23649,8427],{"class":517},[507,23651,625],{"class":583},[507,23653,622],{"class":517},[507,23655,23656],{"class":583},"1.2",[507,23658,622],{"class":517},[507,23660,23661],{"class":583},"1.4",[507,23663,1794],{"class":517},[507,23665,23666,23669,23671,23673,23675,23677,23679,23681,23684,23686,23689],{"class":509,"line":566},[507,23667,23668],{"class":517},"learning_pair_depths ",[507,23670,573],{"class":572},[507,23672,8427],{"class":517},[507,23674,601],{"class":583},[507,23676,622],{"class":517},[507,23678,12152],{"class":583},[507,23680,622],{"class":517},[507,23682,23683],{"class":583},"24",[507,23685,622],{"class":517},[507,23687,23688],{"class":583},"48",[507,23690,1794],{"class":517},[507,23692,23693],{"class":509,"line":590},[507,23694,556],{"emptyLinePlaceholder":133},[507,23696,23697],{"class":509,"line":610},[507,23698,556],{"emptyLinePlaceholder":133},[507,23700,23701,23704,23706],{"class":509,"line":634},[507,23702,23703],{"class":517},"experimental_opts ",[507,23705,573],{"class":572},[507,23707,23708],{"class":517}," {}\n",[507,23710,23711,23714,23717,23719,23721],{"class":509,"line":661},[507,23712,23713],{"class":517},"experimental_opts[",[507,23715,23716],{"class":730},"\"resilience\"",[507,23718,8206],{"class":517},[507,23720,573],{"class":572},[507,23722,23723],{"class":517}," {\n",[507,23725,23726,23729,23731,23733],{"class":509,"line":678},[507,23727,23728],{"class":730},"    \"measure_mitigation\"",[507,23730,1403],{"class":517},[507,23732,13878],{"class":583},[507,23734,1409],{"class":517},[507,23736,23737,23740],{"class":509,"line":683},[507,23738,23739],{"class":730},"    \"measure_noise_learning\"",[507,23741,23742],{"class":517},": {\n",[507,23744,23745,23748],{"class":509,"line":697},[507,23746,23747],{"class":730},"        \"num_randomizations\"",[507,23749,23750],{"class":517},": num_randomizations_learning,\n",[507,23752,23753,23756],{"class":509,"line":710},[507,23754,23755],{"class":730},"        \"shots_per_randomization\"",[507,23757,23758],{"class":517},": shots_per_randomization,\n",[507,23760,23761],{"class":509,"line":715},[507,23762,23763],{"class":517},"    },\n",[507,23765,23766,23769,23771,23773],{"class":509,"line":721},[507,23767,23768],{"class":730},"    \"zne_mitigation\"",[507,23770,1403],{"class":517},[507,23772,13878],{"class":583},[507,23774,1409],{"class":517},[507,23776,23777,23780,23783,23786],{"class":509,"line":736},[507,23778,23779],{"class":730},"    \"zne\"",[507,23781,23782],{"class":517},": {",[507,23784,23785],{"class":730},"\"noise_factors\"",[507,23787,23788],{"class":517},": noise_factors},\n",[507,23790,23791,23794],{"class":509,"line":748},[507,23792,23793],{"class":730},"    \"layer_noise_learning\"",[507,23795,23742],{"class":517},[507,23797,23798,23801,23803,23806],{"class":509,"line":761},[507,23799,23800],{"class":730},"        \"max_layers_to_learn\"",[507,23802,1403],{"class":517},[507,23804,23805],{"class":583},"10",[507,23807,1409],{"class":517},[507,23809,23810,23813],{"class":509,"line":775},[507,23811,23812],{"class":730},"        \"layer_pair_depths\"",[507,23814,23815],{"class":517},": learning_pair_depths,\n",[507,23817,23818,23820],{"class":509,"line":784},[507,23819,23755],{"class":730},[507,23821,23758],{"class":517},[507,23823,23824,23826],{"class":509,"line":796},[507,23825,23747],{"class":730},[507,23827,23750],{"class":517},[507,23829,23830],{"class":509,"line":809},[507,23831,23763],{"class":517},[507,23833,23834,23836],{"class":509,"line":1352},[507,23835,23779],{"class":730},[507,23837,23742],{"class":517},[507,23839,23840,23843,23845,23848],{"class":509,"line":1357},[507,23841,23842],{"class":730},"        \"amplifier\"",[507,23844,1403],{"class":517},[507,23846,23847],{"class":730},"\"pea\"",[507,23849,1409],{"class":517},[507,23851,23852,23855,23858,23860,23862,23864],{"class":509,"line":1362},[507,23853,23854],{"class":730},"        \"extrapolated_noise_factors\"",[507,23856,23857],{"class":517},": [",[507,23859,601],{"class":583},[507,23861,8206],{"class":517},[507,23863,2107],{"class":572},[507,23865,23866],{"class":517}," noise_factors,\n",[507,23868,23869],{"class":509,"line":1367},[507,23870,23763],{"class":517},[507,23872,23873],{"class":509,"line":1379},[507,23874,23875],{"class":517},"}\n",[507,23877,23878,23880,23883,23885,23887],{"class":509,"line":1389},[507,23879,23713],{"class":517},[507,23881,23882],{"class":730},"\"twirling\"",[507,23884,8206],{"class":517},[507,23886,573],{"class":572},[507,23888,23723],{"class":517},[507,23890,23891,23894],{"class":509,"line":1397},[507,23892,23893],{"class":730},"    \"num_randomizations\"",[507,23895,23896],{"class":517},": num_randomizations,\n",[507,23898,23899,23902],{"class":509,"line":1412},[507,23900,23901],{"class":730},"    \"shots_per_randomization\"",[507,23903,23758],{"class":517},[507,23905,23906,23909,23911,23914],{"class":509,"line":1431},[507,23907,23908],{"class":730},"    \"strategy\"",[507,23910,1403],{"class":517},[507,23912,23913],{"class":730},"\"all\"",[507,23915,1409],{"class":517},[507,23917,23918],{"class":509,"line":1449},[507,23919,23875],{"class":517},[507,23921,23922],{"class":509,"line":1465},[507,23923,556],{"emptyLinePlaceholder":133},[507,23925,23926,23929,23931,23934,23936,23939,23941,23944,23947,23949],{"class":509,"line":1471},[507,23927,23928],{"class":517},"estimator ",[507,23930,573],{"class":572},[507,23932,23933],{"class":576}," Estimator",[507,23935,580],{"class":517},[507,23937,23938],{"class":2155},"mode",[507,23940,573],{"class":572},[507,23942,23943],{"class":517},"backend, ",[507,23945,23946],{"class":2155},"options",[507,23948,573],{"class":572},[507,23950,23951],{"class":517},"experimental_opts)\n",[507,23953,23954],{"class":509,"line":1477},[507,23955,556],{"emptyLinePlaceholder":133},[507,23957,23958],{"class":509,"line":1482},[507,23959,556],{"emptyLinePlaceholder":133},[507,23961,23962,23965,23967,23970,23972],{"class":509,"line":1488},[507,23963,23964],{"class":517},"job ",[507,23966,573],{"class":572},[507,23968,23969],{"class":517}," estimator.",[507,23971,22501],{"class":576},[507,23973,23974],{"class":517},"([pub])\n",[13,23976,23978],{"id":23977},"step-4-post-process-and-return-result-in-desired-classical-format","Step 4: Post-process and return result in desired classical format",[498,23980,23982],{"className":500,"code":23981,"language":502,"meta":104,"style":104},"results = job.result()[0]\n",[504,23983,23984],{"__ignoreMap":104},[507,23985,23986,23989,23991,23994,23997,24000,24002],{"class":509,"line":510},[507,23987,23988],{"class":517},"results ",[507,23990,573],{"class":572},[507,23992,23993],{"class":517}," job.",[507,23995,23996],{"class":576},"result",[507,23998,23999],{"class":517},"()[",[507,24001,601],{"class":583},[507,24003,1794],{"class":517},[2513,24005,24007],{"id":24006},"calculate-effective-hamiltonian-and-overlap-matrices","Calculate Effective Hamiltonian and Overlap matrices",[18,24009,24010,24011,24047],{},"First calculate the phase accumulated by the ",[507,24012,24014,24032],{"className":24013},[2523],[507,24015,24017],{"className":24016},[2527],[2529,24018,24019],{"xmlns":2531},[2533,24020,24021,24029],{},[2536,24022,24023,24025,24027],{},[2542,24024,2749],{"mathvariant":2748},[2693,24026,601],{},[2689,24028,2756],{"stretchy":2755},[2549,24030,24031],{"encoding":2551},"\\vert 0 \\rangle",[507,24033,24035],{"className":24034,"ariaHidden":2557},[2556],[507,24036,24038,24041,24044],{"className":24037},[2561],[507,24039],{"className":24040,"style":2769},[2565],[507,24042,9735],{"className":24043},[2570],[507,24045,2756],{"className":24046},[2780]," state during the uncontrolled time evolution",[498,24049,24051],{"className":500,"code":24050,"language":502,"meta":104,"style":104},"prefactors = [\n    np.exp(-1j * sum([c for p, c in H_op.to_list() if \"Z\" in p]) * i * dt)\n    for i in range(1, krylov_dim)\n]\n",[504,24052,24053,24062,24117,24134],{"__ignoreMap":104},[507,24054,24055,24058,24060],{"class":509,"line":510},[507,24056,24057],{"class":517},"prefactors ",[507,24059,573],{"class":572},[507,24061,2177],{"class":517},[507,24063,24064,24067,24070,24072,24074,24076,24078,24080,24082,24085,24087,24090,24092,24094,24096,24098,24100,24103,24105,24108,24110,24112,24114],{"class":509,"line":105},[507,24065,24066],{"class":517},"    np.",[507,24068,24069],{"class":576},"exp",[507,24071,580],{"class":517},[507,24073,2367],{"class":572},[507,24075,625],{"class":583},[507,24077,2372],{"class":513},[507,24079,8229],{"class":572},[507,24081,8815],{"class":572},[507,24083,24084],{"class":517},"([c ",[507,24086,1630],{"class":513},[507,24088,24089],{"class":517}," p, c ",[507,24091,1636],{"class":513},[507,24093,1924],{"class":517},[507,24095,1927],{"class":576},[507,24097,1677],{"class":517},[507,24099,1645],{"class":513},[507,24101,24102],{"class":730}," \"Z\"",[507,24104,21983],{"class":513},[507,24106,24107],{"class":517}," p]) ",[507,24109,2391],{"class":572},[507,24111,8246],{"class":517},[507,24113,2391],{"class":572},[507,24115,24116],{"class":517}," dt)\n",[507,24118,24119,24121,24123,24125,24127,24129,24131],{"class":509,"line":540},[507,24120,1916],{"class":513},[507,24122,8246],{"class":517},[507,24124,1636],{"class":513},[507,24126,8221],{"class":572},[507,24128,580],{"class":517},[507,24130,625],{"class":583},[507,24132,24133],{"class":517},", krylov_dim)\n",[507,24135,24136],{"class":509,"line":553},[507,24137,1794],{"class":517},[18,24139,24140,24141],{},"Once we have the results of the circuit executions we can post-process the data to calculate the matrix elements of ",[507,24142,24144,24157],{"className":24143},[2523],[507,24145,24147],{"className":24146},[2527],[2529,24148,24149],{"xmlns":2531},[2533,24150,24151,24155],{},[2536,24152,24153],{},[2542,24154,5857],{},[2549,24156,5857],{"encoding":2551},[507,24158,24160],{"className":24159,"ariaHidden":2557},[2556],[507,24161,24163,24166],{"className":24162},[2561],[507,24164],{"className":24165,"style":2566},[2565],[507,24167,5857],{"className":24168,"style":5892},[2570,2611],[498,24170,24172],{"className":500,"code":24171,"language":502,"meta":104,"style":104},"# Assemble S, the overlap matrix of dimension D:\nS_first_row = np.zeros(krylov_dim, dtype=complex)\nS_first_row[0] = 1 + 0j\n\n# Add in ancilla-only measurements:\nfor i in range(krylov_dim - 1):\n    # Get expectation values from experiment\n    expval_real = results.data.evs[0][0][\n        i\n    ]  # automatic extrapolated evs if ZNE is used\n    expval_imag = results.data.evs[1][0][\n        i\n    ]  # automatic extrapolated evs if ZNE is used\n\n    # Get expectation values\n    expval = expval_real + 1j * expval_imag\n    S_first_row[i + 1] += prefactors[i] * expval\n\nS_first_row_list = S_first_row.tolist()  # for saving purposes\n\n\nS_circ = np.zeros((krylov_dim, krylov_dim), dtype=complex)\n\n# Distribute entries from first row across matrix:\nfor i, j in it.product(range(krylov_dim), repeat=2):\n    if i >= j:\n        S_circ[j, i] = S_first_row[i - j]\n    else:\n        S_circ[j, i] = np.conj(S_first_row[j - i])\n",[504,24173,24174,24179,24199,24219,24223,24228,24247,24251,24269,24274,24281,24297,24301,24307,24311,24316,24334,24354,24358,24377,24381,24385,24405,24409,24414,24444,24456,24471,24477],{"__ignoreMap":104},[507,24175,24176],{"class":509,"line":510},[507,24177,24178],{"class":562},"# Assemble S, the overlap matrix of dimension D:\n",[507,24180,24181,24184,24186,24188,24190,24193,24195,24197],{"class":509,"line":105},[507,24182,24183],{"class":517},"S_first_row ",[507,24185,573],{"class":572},[507,24187,1616],{"class":517},[507,24189,2149],{"class":576},[507,24191,24192],{"class":517},"(krylov_dim, ",[507,24194,2156],{"class":2155},[507,24196,2159],{"class":572},[507,24198,587],{"class":517},[507,24200,24201,24204,24206,24208,24210,24212,24214,24216],{"class":509,"line":540},[507,24202,24203],{"class":517},"S_first_row[",[507,24205,601],{"class":583},[507,24207,8206],{"class":517},[507,24209,573],{"class":572},[507,24211,1426],{"class":583},[507,24213,8313],{"class":572},[507,24215,21258],{"class":583},[507,24217,24218],{"class":513},"j\n",[507,24220,24221],{"class":509,"line":553},[507,24222,556],{"emptyLinePlaceholder":133},[507,24224,24225],{"class":509,"line":559},[507,24226,24227],{"class":562},"# Add in ancilla-only measurements:\n",[507,24229,24230,24232,24234,24236,24238,24241,24243,24245],{"class":509,"line":566},[507,24231,1630],{"class":513},[507,24233,8246],{"class":517},[507,24235,1636],{"class":513},[507,24237,8221],{"class":572},[507,24239,24240],{"class":517},"(krylov_dim ",[507,24242,2367],{"class":572},[507,24244,1426],{"class":583},[507,24246,1883],{"class":517},[507,24248,24249],{"class":509,"line":590},[507,24250,23359],{"class":562},[507,24252,24253,24255,24257,24260,24262,24264,24266],{"class":509,"line":610},[507,24254,23364],{"class":517},[507,24256,573],{"class":572},[507,24258,24259],{"class":517}," results.data.evs[",[507,24261,601],{"class":583},[507,24263,1755],{"class":517},[507,24265,601],{"class":583},[507,24267,24268],{"class":517},"][\n",[507,24270,24271],{"class":509,"line":634},[507,24272,24273],{"class":517},"        i\n",[507,24275,24276,24278],{"class":509,"line":661},[507,24277,2227],{"class":517},[507,24279,24280],{"class":562},"# automatic extrapolated evs if ZNE is used\n",[507,24282,24283,24285,24287,24289,24291,24293,24295],{"class":509,"line":678},[507,24284,23407],{"class":517},[507,24286,573],{"class":572},[507,24288,24259],{"class":517},[507,24290,625],{"class":583},[507,24292,1755],{"class":517},[507,24294,601],{"class":583},[507,24296,24268],{"class":517},[507,24298,24299],{"class":509,"line":683},[507,24300,24273],{"class":517},[507,24302,24303,24305],{"class":509,"line":697},[507,24304,2227],{"class":517},[507,24306,24280],{"class":562},[507,24308,24309],{"class":509,"line":710},[507,24310,556],{"emptyLinePlaceholder":133},[507,24312,24313],{"class":509,"line":715},[507,24314,24315],{"class":562},"    # Get expectation values\n",[507,24317,24318,24320,24322,24324,24326,24328,24330,24332],{"class":509,"line":721},[507,24319,23448],{"class":517},[507,24321,573],{"class":572},[507,24323,23453],{"class":517},[507,24325,2107],{"class":572},[507,24327,1426],{"class":583},[507,24329,2372],{"class":513},[507,24331,8229],{"class":572},[507,24333,23464],{"class":517},[507,24335,24336,24339,24341,24343,24345,24347,24350,24352],{"class":509,"line":736},[507,24337,24338],{"class":517},"    S_first_row[i ",[507,24340,2107],{"class":572},[507,24342,1426],{"class":583},[507,24344,8206],{"class":517},[507,24346,2285],{"class":572},[507,24348,24349],{"class":517}," prefactors[i] ",[507,24351,2391],{"class":572},[507,24353,23488],{"class":517},[507,24355,24356],{"class":509,"line":748},[507,24357,556],{"emptyLinePlaceholder":133},[507,24359,24360,24363,24365,24368,24371,24374],{"class":509,"line":761},[507,24361,24362],{"class":517},"S_first_row_list ",[507,24364,573],{"class":572},[507,24366,24367],{"class":517}," S_first_row.",[507,24369,24370],{"class":576},"tolist",[507,24372,24373],{"class":517},"()  ",[507,24375,24376],{"class":562},"# for saving purposes\n",[507,24378,24379],{"class":509,"line":775},[507,24380,556],{"emptyLinePlaceholder":133},[507,24382,24383],{"class":509,"line":784},[507,24384,556],{"emptyLinePlaceholder":133},[507,24386,24387,24390,24392,24394,24396,24399,24401,24403],{"class":509,"line":796},[507,24388,24389],{"class":517},"S_circ ",[507,24391,573],{"class":572},[507,24393,1616],{"class":517},[507,24395,2149],{"class":576},[507,24397,24398],{"class":517},"((krylov_dim, krylov_dim), ",[507,24400,2156],{"class":2155},[507,24402,2159],{"class":572},[507,24404,587],{"class":517},[507,24406,24407],{"class":509,"line":809},[507,24408,556],{"emptyLinePlaceholder":133},[507,24410,24411],{"class":509,"line":1352},[507,24412,24413],{"class":562},"# Distribute entries from first row across matrix:\n",[507,24415,24416,24418,24421,24423,24425,24428,24430,24432,24435,24438,24440,24442],{"class":509,"line":1357},[507,24417,1630],{"class":513},[507,24419,24420],{"class":517}," i, j ",[507,24422,1636],{"class":513},[507,24424,2196],{"class":517},[507,24426,24427],{"class":576},"product",[507,24429,580],{"class":517},[507,24431,2204],{"class":572},[507,24433,24434],{"class":517},"(krylov_dim), ",[507,24436,24437],{"class":2155},"repeat",[507,24439,573],{"class":572},[507,24441,584],{"class":583},[507,24443,1883],{"class":517},[507,24445,24446,24448,24450,24453],{"class":509,"line":1362},[507,24447,1717],{"class":513},[507,24449,8246],{"class":517},[507,24451,24452],{"class":572},">=",[507,24454,24455],{"class":517}," j:\n",[507,24457,24458,24461,24463,24466,24468],{"class":509,"line":1367},[507,24459,24460],{"class":517},"        S_circ[j, i] ",[507,24462,573],{"class":572},[507,24464,24465],{"class":517}," S_first_row[i ",[507,24467,2367],{"class":572},[507,24469,24470],{"class":517}," j]\n",[507,24472,24473,24475],{"class":509,"line":1379},[507,24474,1800],{"class":513},[507,24476,1728],{"class":517},[507,24478,24479,24481,24483,24485,24487,24490,24492],{"class":509,"line":1389},[507,24480,24460],{"class":517},[507,24482,573],{"class":572},[507,24484,1616],{"class":517},[507,24486,1674],{"class":576},[507,24488,24489],{"class":517},"(S_first_row[j ",[507,24491,2367],{"class":572},[507,24493,24494],{"class":517}," i])\n",[498,24496,24498],{"className":500,"code":24497,"language":502,"meta":104,"style":104},"Matrix(S_circ)\n",[504,24499,24500],{"__ignoreMap":104},[507,24501,24502,24505],{"class":509,"line":510},[507,24503,24504],{"class":576},"Matrix",[507,24506,24507],{"class":517},"(S_circ)\n",[18,24509,24510,24511],{},"And the matrix elements of ",[507,24512,24514,24531],{"className":24513},[2523],[507,24515,24517],{"className":24516},[2527],[2529,24518,24519],{"xmlns":2531},[2533,24520,24521,24529],{},[2536,24522,24523],{},[4271,24524,24525,24527],{"accent":2557},[2542,24526,3138],{},[2689,24528,4277],{},[2549,24530,4280],{"encoding":2551},[507,24532,24534],{"className":24533,"ariaHidden":2557},[2556],[507,24535,24537,24540],{"className":24536},[2561],[507,24538],{"className":24539,"style":4290},[2565],[507,24541,24543],{"className":24542},[2570,4294],[507,24544,24546],{"className":24545},[2583],[507,24547,24549],{"className":24548},[2587],[507,24550,24552,24560],{"className":24551,"style":4290},[2591],[507,24553,24554,24557],{"style":4306},[507,24555],{"className":24556,"style":4310},[2599],[507,24558,3138],{"className":24559,"style":3153},[2570,2611],[507,24561,24562,24565],{"style":4316},[507,24563],{"className":24564,"style":4310},[2599],[507,24566,24568],{"className":24567,"style":4324},[4323],[507,24569,4277],{"className":24570},[2570],[498,24572,24574],{"className":500,"code":24573,"language":502,"meta":104,"style":104},"# Assemble S, the overlap matrix of dimension D:\nH_first_row = np.zeros(krylov_dim, dtype=complex)\nH_first_row[0] = H_expval\n\nfor obs_idx, (pauli, coeff) in enumerate(zip(H_op.paulis, H_op.coeffs)):\n    # Add in ancilla-only measurements:\n    for i in range(krylov_dim - 1):\n        # Get expectation values from experiment\n        expval_real = results.data.evs[2 + 2 * obs_idx][0][\n            i\n        ]  # automatic extrapolated evs if ZNE is used\n        expval_imag = results.data.evs[2 + 2 * obs_idx + 1][0][\n            i\n        ]  # automatic extrapolated evs if ZNE is used\n\n        # Get expectation values\n        expval = expval_real + 1j * expval_imag\n        H_first_row[i + 1] += prefactors[i] * coeff * expval\n\nH_first_row_list = H_first_row.tolist()\n\nH_eff_circ = np.zeros((krylov_dim, krylov_dim), dtype=complex)\n\n# Distribute entries from first row across matrix:\nfor i, j in it.product(range(krylov_dim), repeat=2):\n    if i >= j:\n        H_eff_circ[j, i] = H_first_row[i - j]\n    else:\n        H_eff_circ[j, i] = np.conj(H_first_row[j - i])\n",[504,24575,24576,24580,24599,24613,24617,24633,24638,24656,24661,24685,24690,24697,24727,24731,24737,24741,24746,24765,24788,24792,24806,24810,24829,24833,24837,24863,24873,24887,24893],{"__ignoreMap":104},[507,24577,24578],{"class":509,"line":510},[507,24579,24178],{"class":562},[507,24581,24582,24585,24587,24589,24591,24593,24595,24597],{"class":509,"line":105},[507,24583,24584],{"class":517},"H_first_row ",[507,24586,573],{"class":572},[507,24588,1616],{"class":517},[507,24590,2149],{"class":576},[507,24592,24192],{"class":517},[507,24594,2156],{"class":2155},[507,24596,2159],{"class":572},[507,24598,587],{"class":517},[507,24600,24601,24604,24606,24608,24610],{"class":509,"line":540},[507,24602,24603],{"class":517},"H_first_row[",[507,24605,601],{"class":583},[507,24607,8206],{"class":517},[507,24609,573],{"class":572},[507,24611,24612],{"class":517}," H_expval\n",[507,24614,24615],{"class":509,"line":553},[507,24616,556],{"emptyLinePlaceholder":133},[507,24618,24619,24621,24623,24625,24627,24629,24631],{"class":509,"line":559},[507,24620,1630],{"class":513},[507,24622,23342],{"class":517},[507,24624,1636],{"class":513},[507,24626,1957],{"class":572},[507,24628,580],{"class":517},[507,24630,23351],{"class":572},[507,24632,23354],{"class":517},[507,24634,24635],{"class":509,"line":566},[507,24636,24637],{"class":562},"    # Add in ancilla-only measurements:\n",[507,24639,24640,24642,24644,24646,24648,24650,24652,24654],{"class":509,"line":590},[507,24641,1916],{"class":513},[507,24643,8246],{"class":517},[507,24645,1636],{"class":513},[507,24647,8221],{"class":572},[507,24649,24240],{"class":517},[507,24651,2367],{"class":572},[507,24653,1426],{"class":583},[507,24655,1883],{"class":517},[507,24657,24658],{"class":509,"line":610},[507,24659,24660],{"class":562},"        # Get expectation values from experiment\n",[507,24662,24663,24666,24668,24670,24672,24674,24676,24678,24681,24683],{"class":509,"line":634},[507,24664,24665],{"class":517},"        expval_real ",[507,24667,573],{"class":572},[507,24669,24259],{"class":517},[507,24671,584],{"class":583},[507,24673,8313],{"class":572},[507,24675,2316],{"class":583},[507,24677,8229],{"class":572},[507,24679,24680],{"class":517}," obs_idx][",[507,24682,601],{"class":583},[507,24684,24268],{"class":517},[507,24686,24687],{"class":509,"line":661},[507,24688,24689],{"class":517},"            i\n",[507,24691,24692,24695],{"class":509,"line":678},[507,24693,24694],{"class":517},"        ]  ",[507,24696,24280],{"class":562},[507,24698,24699,24702,24704,24706,24708,24710,24712,24714,24717,24719,24721,24723,24725],{"class":509,"line":683},[507,24700,24701],{"class":517},"        expval_imag ",[507,24703,573],{"class":572},[507,24705,24259],{"class":517},[507,24707,584],{"class":583},[507,24709,8313],{"class":572},[507,24711,2316],{"class":583},[507,24713,8229],{"class":572},[507,24715,24716],{"class":517}," obs_idx ",[507,24718,2107],{"class":572},[507,24720,1426],{"class":583},[507,24722,1755],{"class":517},[507,24724,601],{"class":583},[507,24726,24268],{"class":517},[507,24728,24729],{"class":509,"line":697},[507,24730,24689],{"class":517},[507,24732,24733,24735],{"class":509,"line":710},[507,24734,24694],{"class":517},[507,24736,24280],{"class":562},[507,24738,24739],{"class":509,"line":715},[507,24740,556],{"emptyLinePlaceholder":133},[507,24742,24743],{"class":509,"line":721},[507,24744,24745],{"class":562},"        # Get expectation values\n",[507,24747,24748,24751,24753,24755,24757,24759,24761,24763],{"class":509,"line":736},[507,24749,24750],{"class":517},"        expval ",[507,24752,573],{"class":572},[507,24754,23453],{"class":517},[507,24756,2107],{"class":572},[507,24758,1426],{"class":583},[507,24760,2372],{"class":513},[507,24762,8229],{"class":572},[507,24764,23464],{"class":517},[507,24766,24767,24770,24772,24774,24776,24778,24780,24782,24784,24786],{"class":509,"line":748},[507,24768,24769],{"class":517},"        H_first_row[i ",[507,24771,2107],{"class":572},[507,24773,1426],{"class":583},[507,24775,8206],{"class":517},[507,24777,2285],{"class":572},[507,24779,24349],{"class":517},[507,24781,2391],{"class":572},[507,24783,23483],{"class":517},[507,24785,2391],{"class":572},[507,24787,23488],{"class":517},[507,24789,24790],{"class":509,"line":761},[507,24791,556],{"emptyLinePlaceholder":133},[507,24793,24794,24797,24799,24802,24804],{"class":509,"line":775},[507,24795,24796],{"class":517},"H_first_row_list ",[507,24798,573],{"class":572},[507,24800,24801],{"class":517}," H_first_row.",[507,24803,24370],{"class":576},[507,24805,781],{"class":517},[507,24807,24808],{"class":509,"line":784},[507,24809,556],{"emptyLinePlaceholder":133},[507,24811,24812,24815,24817,24819,24821,24823,24825,24827],{"class":509,"line":796},[507,24813,24814],{"class":517},"H_eff_circ ",[507,24816,573],{"class":572},[507,24818,1616],{"class":517},[507,24820,2149],{"class":576},[507,24822,24398],{"class":517},[507,24824,2156],{"class":2155},[507,24826,2159],{"class":572},[507,24828,587],{"class":517},[507,24830,24831],{"class":509,"line":809},[507,24832,556],{"emptyLinePlaceholder":133},[507,24834,24835],{"class":509,"line":1352},[507,24836,24413],{"class":562},[507,24838,24839,24841,24843,24845,24847,24849,24851,24853,24855,24857,24859,24861],{"class":509,"line":1357},[507,24840,1630],{"class":513},[507,24842,24420],{"class":517},[507,24844,1636],{"class":513},[507,24846,2196],{"class":517},[507,24848,24427],{"class":576},[507,24850,580],{"class":517},[507,24852,2204],{"class":572},[507,24854,24434],{"class":517},[507,24856,24437],{"class":2155},[507,24858,573],{"class":572},[507,24860,584],{"class":583},[507,24862,1883],{"class":517},[507,24864,24865,24867,24869,24871],{"class":509,"line":1362},[507,24866,1717],{"class":513},[507,24868,8246],{"class":517},[507,24870,24452],{"class":572},[507,24872,24455],{"class":517},[507,24874,24875,24878,24880,24883,24885],{"class":509,"line":1367},[507,24876,24877],{"class":517},"        H_eff_circ[j, i] ",[507,24879,573],{"class":572},[507,24881,24882],{"class":517}," H_first_row[i ",[507,24884,2367],{"class":572},[507,24886,24470],{"class":517},[507,24888,24889,24891],{"class":509,"line":1379},[507,24890,1800],{"class":513},[507,24892,1728],{"class":517},[507,24894,24895,24897,24899,24901,24903,24906,24908],{"class":509,"line":1389},[507,24896,24877],{"class":517},[507,24898,573],{"class":572},[507,24900,1616],{"class":517},[507,24902,1674],{"class":576},[507,24904,24905],{"class":517},"(H_first_row[j ",[507,24907,2367],{"class":572},[507,24909,24494],{"class":517},[498,24911,24913],{"className":500,"code":24912,"language":502,"meta":104,"style":104},"Matrix(H_eff_circ)\n",[504,24914,24915],{"__ignoreMap":104},[507,24916,24917,24919],{"class":509,"line":510},[507,24918,24504],{"class":576},[507,24920,24921],{"class":517},"(H_eff_circ)\n",[18,24923,24924,24925,24985],{},"Finally, we can solve the generalized eigenvalue problem for ",[507,24926,24928,24945],{"className":24927},[2523],[507,24929,24931],{"className":24930},[2527],[2529,24932,24933],{"xmlns":2531},[2533,24934,24935,24943],{},[2536,24936,24937],{},[4271,24938,24939,24941],{"accent":2557},[2542,24940,3138],{},[2689,24942,4277],{},[2549,24944,4280],{"encoding":2551},[507,24946,24948],{"className":24947,"ariaHidden":2557},[2556],[507,24949,24951,24954],{"className":24950},[2561],[507,24952],{"className":24953,"style":4290},[2565],[507,24955,24957],{"className":24956},[2570,4294],[507,24958,24960],{"className":24959},[2583],[507,24961,24963],{"className":24962},[2587],[507,24964,24966,24974],{"className":24965,"style":4290},[2591],[507,24967,24968,24971],{"style":4306},[507,24969],{"className":24970,"style":4310},[2599],[507,24972,3138],{"className":24973,"style":3153},[2570,2611],[507,24975,24976,24979],{"style":4316},[507,24977],{"className":24978,"style":4310},[2599],[507,24980,24982],{"className":24981,"style":4324},[4323],[507,24983,4277],{"className":24984},[2570],":",[18,24987,24988],{},[507,24989,24991,25027],{"className":24990},[2523],[507,24992,24994],{"className":24993},[2527],[2529,24995,24996],{"xmlns":2531},[2533,24997,24998,25024],{},[2536,24999,25000,25006,25012,25014,25016,25018],{},[4271,25001,25002,25004],{"accent":2557},[2542,25003,3138],{},[2689,25005,4277],{},[4271,25007,25008,25010],{"accent":2557},[2542,25009,6174],{},[2689,25011,6177],{},[2689,25013,573],{},[2542,25015,6174],{},[2542,25017,5857],{},[4271,25019,25020,25022],{"accent":2557},[2542,25021,6174],{},[2689,25023,6177],{},[2549,25025,25026],{"encoding":2551},"\\tilde{H} \\vec{c} = c S \\vec{c}",[507,25028,25030,25111],{"className":25029,"ariaHidden":2557},[2556],[507,25031,25033,25036,25067,25102,25105,25108],{"className":25032},[2561],[507,25034],{"className":25035,"style":4290},[2565],[507,25037,25039],{"className":25038},[2570,4294],[507,25040,25042],{"className":25041},[2583],[507,25043,25045],{"className":25044},[2587],[507,25046,25048,25056],{"className":25047,"style":4290},[2591],[507,25049,25050,25053],{"style":4306},[507,25051],{"className":25052,"style":4310},[2599],[507,25054,3138],{"className":25055,"style":3153},[2570,2611],[507,25057,25058,25061],{"style":4316},[507,25059],{"className":25060,"style":4310},[2599],[507,25062,25064],{"className":25063,"style":4324},[4323],[507,25065,4277],{"className":25066},[2570],[507,25068,25070],{"className":25069},[2570,4294],[507,25071,25073],{"className":25072},[2583],[507,25074,25076],{"className":25075},[2587],[507,25077,25079,25087],{"className":25078,"style":6257},[2591],[507,25080,25081,25084],{"style":4306},[507,25082],{"className":25083,"style":4310},[2599],[507,25085,6174],{"className":25086},[2570,2611],[507,25088,25089,25092],{"style":4306},[507,25090],{"className":25091,"style":4310},[2599],[507,25093,25095],{"className":25094,"style":6274},[4323],[507,25096,25098],{"className":25097,"style":6279},[6278],[6281,25099,25100],{"xmlns":6283,"width":6284,"height":6285,"style":6286,"viewBox":6287,"preserveAspectRatio":6288},[6290,25101],{"d":6292},[507,25103],{"className":25104,"style":2919},[2714],[507,25106,573],{"className":25107},[2923],[507,25109],{"className":25110,"style":2919},[2714],[507,25112,25114,25117,25120,25123],{"className":25113},[2561],[507,25115],{"className":25116,"style":6257},[2565],[507,25118,6174],{"className":25119},[2570,2611],[507,25121,5857],{"className":25122,"style":5892},[2570,2611],[507,25124,25126],{"className":25125},[2570,4294],[507,25127,25129],{"className":25128},[2583],[507,25130,25132],{"className":25131},[2587],[507,25133,25135,25143],{"className":25134,"style":6257},[2591],[507,25136,25137,25140],{"style":4306},[507,25138],{"className":25139,"style":4310},[2599],[507,25141,6174],{"className":25142},[2570,2611],[507,25144,25145,25148],{"style":4306},[507,25146],{"className":25147,"style":4310},[2599],[507,25149,25151],{"className":25150,"style":6274},[4323],[507,25152,25154],{"className":25153,"style":6279},[6278],[6281,25155,25156],{"xmlns":6283,"width":6284,"height":6285,"style":6286,"viewBox":6287,"preserveAspectRatio":6288},[6290,25157],{"d":6292},[18,25159,25160,25161],{},"and get an estimate of the ground state energy ",[507,25162,25164,25188],{"className":25163},[2523],[507,25165,25167],{"className":25166},[2527],[2529,25168,25169],{"xmlns":2531},[2533,25170,25171,25185],{},[2536,25172,25173],{},[3168,25174,25175,25177],{},[2542,25176,6174],{},[2536,25178,25179,25181,25183],{},[2542,25180,4417],{},[2542,25182,3293],{},[2542,25184,4420],{},[2549,25186,25187],{"encoding":2551},"c_{min}",[507,25189,25191],{"className":25190,"ariaHidden":2557},[2556],[507,25192,25194,25198],{"className":25193},[2561],[507,25195],{"className":25196,"style":25197},[2565],"height:0.5806em;vertical-align:-0.15em;",[507,25199,25201,25204],{"className":25200},[2570],[507,25202,6174],{"className":25203},[2570,2611],[507,25205,25207],{"className":25206},[2579],[507,25208,25210,25234],{"className":25209},[2583,3200],[507,25211,25213,25231],{"className":25212},[2587],[507,25214,25216],{"className":25215,"style":6802},[2591],[507,25217,25218,25221],{"style":5398},[507,25219],{"className":25220,"style":2600},[2599],[507,25222,25224],{"className":25223},[2604,2605,2606,2607],[507,25225,25227],{"className":25226},[2570,2607],[507,25228,25230],{"className":25229},[2570,2611,2607],"min",[507,25232,3225],{"className":25233},[3224],[507,25235,25237],{"className":25236},[2587],[507,25238,25240],{"className":25239,"style":3232},[2591],[507,25241],{},[498,25243,25245],{"className":500,"code":25244,"language":502,"meta":104,"style":104},"gnd_en_circ_est_list = []\nfor d in range(1, krylov_dim + 1):\n    # Solve generalized eigenvalue problem for different size of the Krylov space\n    gnd_en_circ_est = solve_regularized_gen_eig(\n        H_eff_circ[:d, :d], S_circ[:d, :d], threshold=9e-1\n    )\n    gnd_en_circ_est_list.append(gnd_en_circ_est)\n    print(\"The estimated ground state energy is: \", gnd_en_circ_est)\n",[504,25246,25247,25256,25280,25285,25296,25309,25313,25323],{"__ignoreMap":104},[507,25248,25249,25252,25254],{"class":509,"line":510},[507,25250,25251],{"class":517},"gnd_en_circ_est_list ",[507,25253,573],{"class":572},[507,25255,1910],{"class":517},[507,25257,25258,25260,25263,25265,25267,25269,25271,25274,25276,25278],{"class":509,"line":105},[507,25259,1630],{"class":513},[507,25261,25262],{"class":517}," d ",[507,25264,1636],{"class":513},[507,25266,8221],{"class":572},[507,25268,580],{"class":517},[507,25270,625],{"class":583},[507,25272,25273],{"class":517},", krylov_dim ",[507,25275,2107],{"class":572},[507,25277,1426],{"class":583},[507,25279,1883],{"class":517},[507,25281,25282],{"class":509,"line":540},[507,25283,25284],{"class":562},"    # Solve generalized eigenvalue problem for different size of the Krylov space\n",[507,25286,25287,25290,25292,25294],{"class":509,"line":553},[507,25288,25289],{"class":517},"    gnd_en_circ_est ",[507,25291,573],{"class":572},[507,25293,1373],{"class":576},[507,25295,1376],{"class":517},[507,25297,25298,25301,25304,25306],{"class":509,"line":559},[507,25299,25300],{"class":517},"        H_eff_circ[:d, :d], S_circ[:d, :d], ",[507,25302,25303],{"class":2155},"threshold",[507,25305,573],{"class":572},[507,25307,25308],{"class":583},"9e-1\n",[507,25310,25311],{"class":509,"line":566},[507,25312,1660],{"class":517},[507,25314,25315,25318,25320],{"class":509,"line":590},[507,25316,25317],{"class":517},"    gnd_en_circ_est_list.",[507,25319,1939],{"class":576},[507,25321,25322],{"class":517},"(gnd_en_circ_est)\n",[507,25324,25325,25327,25329,25332],{"class":509,"line":610},[507,25326,2060],{"class":572},[507,25328,580],{"class":517},[507,25330,25331],{"class":730},"\"The estimated ground state energy is: \"",[507,25333,25334],{"class":517},", gnd_en_circ_est)\n",[498,25336,25339],{"className":25337,"code":25338,"language":7039,"meta":104},[8531],"The estimated ground state energy is:  25.0\nThe estimated ground state energy is:  22.572154819954875\nThe estimated ground state energy is:  21.691509219286587\nThe estimated ground state energy is:  21.23882298756386\nThe estimated ground state energy is:  20.965499325470294\n",[504,25340,25338],{"__ignoreMap":104},[18,25342,25343],{},"For a single-particle sector, we can efficiently calculate the ground state of this sector of the Hamiltonian classically",[498,25345,25347],{"className":500,"code":25346,"language":502,"meta":104,"style":104},"gs_en = single_particle_gs(H_op, n_qubits)\n",[504,25348,25349],{"__ignoreMap":104},[507,25350,25351,25354,25356,25358],{"class":509,"line":510},[507,25352,25353],{"class":517},"gs_en ",[507,25355,573],{"class":572},[507,25357,1870],{"class":576},[507,25359,25360],{"class":517},"(H_op, n_qubits)\n",[498,25362,25365],{"className":25363,"code":25364,"language":7039,"meta":104},[8531],"n_sys_qubits 30\nn_exc 1 , subspace dimension 31\nsingle particle ground state energy:  21.021912418526906\n",[504,25366,25364],{"__ignoreMap":104},[498,25368,25370],{"className":500,"code":25369,"language":502,"meta":104,"style":104},"plt.plot(\n    range(1, krylov_dim + 1),\n    gnd_en_circ_est_list,\n    color=\"blue\",\n    linestyle=\"-.\",\n    label=\"KQD estimate\",\n)\nplt.plot(\n    range(1, krylov_dim + 1),\n    [gs_en] * krylov_dim,\n    color=\"red\",\n    linestyle=\"-\",\n    label=\"exact\",\n)\nplt.xticks(range(1, krylov_dim + 1), range(1, krylov_dim + 1))\nplt.legend()\nplt.xlabel(\"Krylov space dimension\")\nplt.ylabel(\"Energy\")\nplt.title(\n    \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n)\nplt.show()\n",[504,25371,25372,25382,25399,25404,25416,25428,25440,25444,25452,25468,25478,25489,25500,25511,25515,25552,25561,25575,25589,25598,25603,25607],{"__ignoreMap":104},[507,25373,25374,25377,25380],{"class":509,"line":510},[507,25375,25376],{"class":517},"plt.",[507,25378,25379],{"class":576},"plot",[507,25381,1376],{"class":517},[507,25383,25384,25387,25389,25391,25393,25395,25397],{"class":509,"line":105},[507,25385,25386],{"class":572},"    range",[507,25388,580],{"class":517},[507,25390,625],{"class":583},[507,25392,25273],{"class":517},[507,25394,2107],{"class":572},[507,25396,1426],{"class":583},[507,25398,14202],{"class":517},[507,25400,25401],{"class":509,"line":540},[507,25402,25403],{"class":517},"    gnd_en_circ_est_list,\n",[507,25405,25406,25409,25411,25414],{"class":509,"line":553},[507,25407,25408],{"class":2155},"    color",[507,25410,573],{"class":572},[507,25412,25413],{"class":730},"\"blue\"",[507,25415,1409],{"class":517},[507,25417,25418,25421,25423,25426],{"class":509,"line":559},[507,25419,25420],{"class":2155},"    linestyle",[507,25422,573],{"class":572},[507,25424,25425],{"class":730},"\"-.\"",[507,25427,1409],{"class":517},[507,25429,25430,25433,25435,25438],{"class":509,"line":566},[507,25431,25432],{"class":2155},"    label",[507,25434,573],{"class":572},[507,25436,25437],{"class":730},"\"KQD estimate\"",[507,25439,1409],{"class":517},[507,25441,25442],{"class":509,"line":590},[507,25443,587],{"class":517},[507,25445,25446,25448,25450],{"class":509,"line":610},[507,25447,25376],{"class":517},[507,25449,25379],{"class":576},[507,25451,1376],{"class":517},[507,25453,25454,25456,25458,25460,25462,25464,25466],{"class":509,"line":634},[507,25455,25386],{"class":572},[507,25457,580],{"class":517},[507,25459,625],{"class":583},[507,25461,25273],{"class":517},[507,25463,2107],{"class":572},[507,25465,1426],{"class":583},[507,25467,14202],{"class":517},[507,25469,25470,25473,25475],{"class":509,"line":661},[507,25471,25472],{"class":517},"    [gs_en] ",[507,25474,2391],{"class":572},[507,25476,25477],{"class":517}," krylov_dim,\n",[507,25479,25480,25482,25484,25487],{"class":509,"line":678},[507,25481,25408],{"class":2155},[507,25483,573],{"class":572},[507,25485,25486],{"class":730},"\"red\"",[507,25488,1409],{"class":517},[507,25490,25491,25493,25495,25498],{"class":509,"line":683},[507,25492,25420],{"class":2155},[507,25494,573],{"class":572},[507,25496,25497],{"class":730},"\"-\"",[507,25499,1409],{"class":517},[507,25501,25502,25504,25506,25509],{"class":509,"line":697},[507,25503,25432],{"class":2155},[507,25505,573],{"class":572},[507,25507,25508],{"class":730},"\"exact\"",[507,25510,1409],{"class":517},[507,25512,25513],{"class":509,"line":710},[507,25514,587],{"class":517},[507,25516,25517,25519,25522,25524,25526,25528,25530,25532,25534,25536,25538,25540,25542,25544,25546,25548,25550],{"class":509,"line":715},[507,25518,25376],{"class":517},[507,25520,25521],{"class":576},"xticks",[507,25523,580],{"class":517},[507,25525,2204],{"class":572},[507,25527,580],{"class":517},[507,25529,625],{"class":583},[507,25531,25273],{"class":517},[507,25533,2107],{"class":572},[507,25535,1426],{"class":583},[507,25537,2213],{"class":517},[507,25539,2204],{"class":572},[507,25541,580],{"class":517},[507,25543,625],{"class":583},[507,25545,25273],{"class":517},[507,25547,2107],{"class":572},[507,25549,1426],{"class":583},[507,25551,22540],{"class":517},[507,25553,25554,25556,25559],{"class":509,"line":721},[507,25555,25376],{"class":517},[507,25557,25558],{"class":576},"legend",[507,25560,781],{"class":517},[507,25562,25563,25565,25568,25570,25573],{"class":509,"line":736},[507,25564,25376],{"class":517},[507,25566,25567],{"class":576},"xlabel",[507,25569,580],{"class":517},[507,25571,25572],{"class":730},"\"Krylov space dimension\"",[507,25574,587],{"class":517},[507,25576,25577,25579,25582,25584,25587],{"class":509,"line":748},[507,25578,25376],{"class":517},[507,25580,25581],{"class":576},"ylabel",[507,25583,580],{"class":517},[507,25585,25586],{"class":730},"\"Energy\"",[507,25588,587],{"class":517},[507,25590,25591,25593,25596],{"class":509,"line":761},[507,25592,25376],{"class":517},[507,25594,25595],{"class":576},"title",[507,25597,1376],{"class":517},[507,25599,25600],{"class":509,"line":775},[507,25601,25602],{"class":730},"    \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n",[507,25604,25605],{"class":509,"line":784},[507,25606,587],{"class":517},[507,25608,25609,25611,25614],{"class":509,"line":796},[507,25610,25376],{"class":517},[507,25612,25613],{"class":576},"show",[507,25615,781],{"class":517},[13,25617,25619],{"id":25618},"appendix-krylov-subspace-from-real-time-evolutions","Appendix: Krylov subspace from real time-evolutions",[18,25621,25622],{},"The unitary Krylov space is defined as",[507,25624,25626],{"className":25625},[2784],[507,25627,25629,25737],{"className":25628},[2523],[507,25630,25632],{"className":25631},[2527],[2529,25633,25634],{"xmlns":2531,"display":2793},[2533,25635,25636,25734],{},[2536,25637,25638,25644,25646,25648,25650,25652,25654,25656,25658,25660,25662],{},[3168,25639,25640,25642],{},[2542,25641,2545],{"mathvariant":2544},[2542,25643,3279],{},[2689,25645,580],{"stretchy":2755},[2542,25647,3138],{},[2689,25649,2819],{"separator":2557},[2542,25651,2749],{"mathvariant":2748},[2542,25653,3793],{},[2689,25655,2756],{"stretchy":2755},[2689,25657,3649],{"stretchy":2755},[2689,25659,573],{},[6167,25661,507],{},[2536,25663,25664,25666,25668,25670,25672,25674,25693,25695,25697,25699,25701,25704,25706,25726,25728,25730,25732],{},[2689,25665,2810],{"fence":2557},[2542,25667,2749],{"mathvariant":2748},[2542,25669,3793],{},[2689,25671,2756],{"stretchy":2755},[2689,25673,2819],{"separator":2557},[2539,25675,25676,25678],{},[2542,25677,3286],{},[2536,25679,25680,25682,25684,25686,25689,25691],{},[2689,25681,2691],{},[2542,25683,3293],{},[2542,25685,3138],{},[6167,25687,25688],{}," ",[2542,25690,4959],{},[2542,25692,3298],{},[2542,25694,2749],{"mathvariant":2748},[2542,25696,3793],{},[2689,25698,2756],{"stretchy":2755},[2689,25700,2819],{"separator":2557},[2689,25702,25703],{},"…",[2689,25705,2819],{"separator":2557},[2539,25707,25708,25710],{},[2542,25709,3286],{},[2536,25711,25712,25714,25716,25718,25720,25722,25724],{},[2689,25713,2691],{},[2542,25715,3293],{},[2542,25717,2216],{},[2542,25719,3138],{},[6167,25721,25688],{},[2542,25723,4959],{},[2542,25725,3298],{},[2542,25727,2749],{"mathvariant":2748},[2542,25729,3793],{},[2689,25731,2756],{"stretchy":2755},[2689,25733,2872],{"fence":2557},[2549,25735,25736],{"encoding":2551},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ |\\psi\\rangle,  e^{-iH\\,dt} |\\psi\\rangle, \\dots, e^{-irH\\,dt} |\\psi\\rangle \\right\\}",[507,25738,25740,25817],{"className":25739,"ariaHidden":2557},[2556],[507,25741,25743,25746,25786,25789,25792,25795,25798,25801,25804,25808,25811,25814],{"className":25742},[2561],[507,25744],{"className":25745,"style":2769},[2565],[507,25747,25749,25752],{"className":25748},[2570],[507,25750,2545],{"className":25751,"style":2575},[2570,2574],[507,25753,25755],{"className":25754},[2579],[507,25756,25758,25778],{"className":25757},[2583,3200],[507,25759,25761,25775],{"className":25760},[2587],[507,25762,25764],{"className":25763,"style":3207},[2591],[507,25765,25766,25769],{"style":3210},[507,25767],{"className":25768,"style":2600},[2599],[507,25770,25772],{"className":25771},[2604,2605,2606,2607],[507,25773,3279],{"className":25774,"style":3314},[2570,2611,2607],[507,25776,3225],{"className":25777},[3224],[507,25779,25781],{"className":25780},[2587],[507,25782,25784],{"className":25783,"style":3232},[2591],[507,25785],{},[507,25787,580],{"className":25788},[2941],[507,25790,3138],{"className":25791,"style":3153},[2570,2611],[507,25793,2819],{"className":25794},[2961],[507,25796],{"className":25797,"style":2965},[2714],[507,25799,2749],{"className":25800},[2570],[507,25802,3793],{"className":25803,"style":2776},[2570,2611],[507,25805,25807],{"className":25806},[2780],"⟩)",[507,25809],{"className":25810,"style":2919},[2714],[507,25812,573],{"className":25813},[2923],[507,25815],{"className":25816,"style":2919},[2714],[507,25818,25820,25824,25830,25833],{"className":25819},[2561],[507,25821],{"className":25822,"style":25823},[2565],"height:1.2491em;vertical-align:-0.35em;",[507,25825,25827],{"className":25826},[2570,7039],[507,25828,507],{"className":25829},[2570],[507,25831],{"className":25832,"style":2965},[2714],[507,25834,25836,25842,25845,25848,25851,25854,25857,25905,25908,25911,25914,25917,25920,25923,25926,25929,25932,25982,25985,25988,25991],{"className":25835},[2937],[507,25837,25839],{"className":25838,"style":2943},[2941,2942],[507,25840,2810],{"className":25841},[2947,2948],[507,25843,2749],{"className":25844},[2570],[507,25846,3793],{"className":25847,"style":2776},[2570,2611],[507,25849,2756],{"className":25850},[2780],[507,25852,2819],{"className":25853},[2961],[507,25855],{"className":25856,"style":2965},[2714],[507,25858,25860,25863],{"className":25859},[2570],[507,25861,3286],{"className":25862},[2570,2611],[507,25864,25866],{"className":25865},[2579],[507,25867,25869],{"className":25868},[2583],[507,25870,25872],{"className":25871},[2587],[507,25873,25875],{"className":25874,"style":5040},[2591],[507,25876,25877,25880],{"style":2906},[507,25878],{"className":25879,"style":2600},[2599],[507,25881,25883],{"className":25882},[2604,2605,2606,2607],[507,25884,25886,25889,25892,25895,25899,25902],{"className":25885},[2570,2607],[507,25887,2691],{"className":25888},[2570,2607],[507,25890,3293],{"className":25891},[2570,2611,2607],[507,25893,3138],{"className":25894,"style":3153},[2570,2611,2607],[507,25896],{"className":25897,"style":25898},[2714,2607],"margin-right:0.1952em;",[507,25900,4959],{"className":25901},[2570,2611,2607],[507,25903,3298],{"className":25904},[2570,2611,2607],[507,25906,2749],{"className":25907},[2570],[507,25909,3793],{"className":25910,"style":2776},[2570,2611],[507,25912,2756],{"className":25913},[2780],[507,25915,2819],{"className":25916},[2961],[507,25918],{"className":25919,"style":2965},[2714],[507,25921,25703],{"className":25922},[2937],[507,25924],{"className":25925,"style":2965},[2714],[507,25927,2819],{"className":25928},[2961],[507,25930],{"className":25931,"style":2965},[2714],[507,25933,25935,25938],{"className":25934},[2570],[507,25936,3286],{"className":25937},[2570,2611],[507,25939,25941],{"className":25940},[2579],[507,25942,25944],{"className":25943},[2583],[507,25945,25947],{"className":25946},[2587],[507,25948,25950],{"className":25949,"style":5040},[2591],[507,25951,25952,25955],{"style":2906},[507,25953],{"className":25954,"style":2600},[2599],[507,25956,25958],{"className":25957},[2604,2605,2606,2607],[507,25959,25961,25964,25967,25970,25973,25976,25979],{"className":25960},[2570,2607],[507,25962,2691],{"className":25963},[2570,2607],[507,25965,3293],{"className":25966},[2570,2611,2607],[507,25968,2216],{"className":25969,"style":2612},[2570,2611,2607],[507,25971,3138],{"className":25972,"style":3153},[2570,2611,2607],[507,25974],{"className":25975,"style":25898},[2714,2607],[507,25977,4959],{"className":25978},[2570,2611,2607],[507,25980,3298],{"className":25981},[2570,2611,2607],[507,25983,2749],{"className":25984},[2570],[507,25986,3793],{"className":25987,"style":2776},[2570,2611],[507,25989,2756],{"className":25990},[2780],[507,25992,25994],{"className":25993,"style":2943},[2780,2942],[507,25995,2872],{"className":25996},[2947,2948],[18,25998,25999,26000,26033,26034,26062,26063,26122],{},"for some timestep ",[507,26001,26003,26018],{"className":26002},[2523],[507,26004,26006],{"className":26005},[2527],[2529,26007,26008],{"xmlns":2531},[2533,26009,26010,26016],{},[2536,26011,26012,26014],{},[2542,26013,4959],{},[2542,26015,3298],{},[2549,26017,5463],{"encoding":2551},[507,26019,26021],{"className":26020,"ariaHidden":2557},[2556],[507,26022,26024,26027,26030],{"className":26023},[2561],[507,26025],{"className":26026,"style":5434},[2565],[507,26028,4959],{"className":26029},[2570,2611],[507,26031,3298],{"className":26032},[2570,2611]," that we will determine later. Temporarily assume ",[507,26035,26037,26050],{"className":26036},[2523],[507,26038,26040],{"className":26039},[2527],[2529,26041,26042],{"xmlns":2531},[2533,26043,26044,26048],{},[2536,26045,26046],{},[2542,26047,2216],{},[2549,26049,2216],{"encoding":2551},[507,26051,26053],{"className":26052,"ariaHidden":2557},[2556],[507,26054,26056,26059],{"className":26055},[2561],[507,26057],{"className":26058,"style":2639},[2565],[507,26060,2216],{"className":26061,"style":2612},[2570,2611]," is even: then define ",[507,26064,26066,26088],{"className":26065},[2523],[507,26067,26069],{"className":26068},[2527],[2529,26070,26071],{"xmlns":2531},[2533,26072,26073,26085],{},[2536,26074,26075,26077,26079,26081,26083],{},[2542,26076,4959],{},[2689,26078,573],{},[2542,26080,2216],{},[2542,26082,645],{"mathvariant":2748},[2693,26084,584],{},[2549,26086,26087],{"encoding":2551},"d=r\u002F2",[507,26089,26091,26109],{"className":26090,"ariaHidden":2557},[2556],[507,26092,26094,26097,26100,26103,26106],{"className":26093},[2561],[507,26095],{"className":26096,"style":5434},[2565],[507,26098,4959],{"className":26099},[2570,2611],[507,26101],{"className":26102,"style":2919},[2714],[507,26104,573],{"className":26105},[2923],[507,26107],{"className":26108,"style":2919},[2714],[507,26110,26112,26115,26118],{"className":26111},[2561],[507,26113],{"className":26114,"style":2769},[2565],[507,26116,2216],{"className":26117,"style":2612},[2570,2611],[507,26119,26121],{"className":26120},[2570],"\u002F2",". Notice that when we project the Hamiltonian into the Krylov space above, it is indistinguishable from the Krylov space",[507,26124,26126],{"className":26125},[2784],[507,26127,26129,26307],{"className":26128},[2523],[507,26130,26132],{"className":26131},[2527],[2529,26133,26134],{"xmlns":2531,"display":2793},[2533,26135,26136,26304],{},[2536,26137,26138,26144,26146,26148,26150,26152,26154,26156,26158,26160,26162,26302],{},[3168,26139,26140,26142],{},[2542,26141,2545],{"mathvariant":2544},[2542,26143,3279],{},[2689,26145,580],{"stretchy":2755},[2542,26147,3138],{},[2689,26149,2819],{"separator":2557},[2542,26151,2749],{"mathvariant":2748},[2542,26153,3793],{},[2689,26155,2756],{"stretchy":2755},[2689,26157,3649],{"stretchy":2755},[2689,26159,573],{},[6167,26161,507],{},[2536,26163,26164,26166,26188,26190,26192,26194,26196,26222,26224,26226,26228,26230,26232,26234,26262,26264,26266,26268,26270,26294,26296,26298,26300],{},[2689,26165,2810],{"fence":2557},[2539,26167,26168,26170],{},[2542,26169,3286],{},[2536,26171,26172,26174,26176,26178,26180,26182,26184,26186],{},[2542,26173,3293],{},[6167,26175,25688],{},[2542,26177,4959],{},[6167,26179,25688],{},[2542,26181,3138],{},[6167,26183,25688],{},[2542,26185,4959],{},[2542,26187,3298],{},[2542,26189,2749],{"mathvariant":2748},[2542,26191,3793],{},[2689,26193,2756],{"stretchy":2755},[2689,26195,2819],{"separator":2557},[2539,26197,26198,26200],{},[2542,26199,3286],{},[2536,26201,26202,26204,26206,26208,26210,26212,26214,26216,26218,26220],{},[2542,26203,3293],{},[2689,26205,580],{"stretchy":2755},[2542,26207,4959],{},[2689,26209,2691],{},[2693,26211,625],{},[2689,26213,3649],{"stretchy":2755},[2542,26215,3138],{},[6167,26217,25688],{},[2542,26219,4959],{},[2542,26221,3298],{},[2542,26223,2749],{"mathvariant":2748},[2542,26225,3793],{},[2689,26227,2756],{"stretchy":2755},[2689,26229,2819],{"separator":2557},[2689,26231,25703],{},[2689,26233,2819],{"separator":2557},[2539,26235,26236,26238],{},[2542,26237,3286],{},[2536,26239,26240,26242,26244,26246,26248,26250,26252,26254,26256,26258,26260],{},[2689,26241,2691],{},[2542,26243,3293],{},[2689,26245,580],{"stretchy":2755},[2542,26247,4959],{},[2689,26249,2691],{},[2693,26251,625],{},[2689,26253,3649],{"stretchy":2755},[2542,26255,3138],{},[6167,26257,25688],{},[2542,26259,4959],{},[2542,26261,3298],{},[2542,26263,2749],{"mathvariant":2748},[2542,26265,3793],{},[2689,26267,2756],{"stretchy":2755},[2689,26269,2819],{"separator":2557},[2539,26271,26272,26274],{},[2542,26273,3286],{},[2536,26275,26276,26278,26280,26282,26284,26286,26288,26290,26292],{},[2689,26277,2691],{},[2542,26279,3293],{},[6167,26281,25688],{},[2542,26283,4959],{},[6167,26285,25688],{},[2542,26287,3138],{},[6167,26289,25688],{},[2542,26291,4959],{},[2542,26293,3298],{},[2542,26295,2749],{"mathvariant":2748},[2542,26297,3793],{},[2689,26299,2756],{"stretchy":2755},[2689,26301,2872],{"fence":2557},[2689,26303,2819],{"separator":2557},[2549,26305,26306],{"encoding":2551},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ e^{i\\,d\\,H\\,dt}|\\psi\\rangle,  e^{i(d-1)H\\,dt} |\\psi\\rangle, \\dots, e^{-i(d-1)H\\,dt} |\\psi\\rangle, e^{-i\\,d\\,H\\,dt} |\\psi\\rangle \\right\\},",[507,26308,26310,26386],{"className":26309,"ariaHidden":2557},[2556],[507,26311,26313,26316,26356,26359,26362,26365,26368,26371,26374,26377,26380,26383],{"className":26312},[2561],[507,26314],{"className":26315,"style":2769},[2565],[507,26317,26319,26322],{"className":26318},[2570],[507,26320,2545],{"className":26321,"style":2575},[2570,2574],[507,26323,26325],{"className":26324},[2579],[507,26326,26328,26348],{"className":26327},[2583,3200],[507,26329,26331,26345],{"className":26330},[2587],[507,26332,26334],{"className":26333,"style":3207},[2591],[507,26335,26336,26339],{"style":3210},[507,26337],{"className":26338,"style":2600},[2599],[507,26340,26342],{"className":26341},[2604,2605,2606,2607],[507,26343,3279],{"className":26344,"style":3314},[2570,2611,2607],[507,26346,3225],{"className":26347},[3224],[507,26349,26351],{"className":26350},[2587],[507,26352,26354],{"className":26353,"style":3232},[2591],[507,26355],{},[507,26357,580],{"className":26358},[2941],[507,26360,3138],{"className":26361,"style":3153},[2570,2611],[507,26363,2819],{"className":26364},[2961],[507,26366],{"className":26367,"style":2965},[2714],[507,26369,2749],{"className":26370},[2570],[507,26372,3793],{"className":26373,"style":2776},[2570,2611],[507,26375,25807],{"className":26376},[2780],[507,26378],{"className":26379,"style":2919},[2714],[507,26381,573],{"className":26382},[2923],[507,26384],{"className":26385,"style":2919},[2714],[507,26387,26389,26392,26398,26401,26713,26716],{"className":26388},[2561],[507,26390],{"className":26391,"style":13545},[2565],[507,26393,26395],{"className":26394},[2570,7039],[507,26396,507],{"className":26397},[2570],[507,26399],{"className":26400,"style":2965},[2714],[507,26402,26404,26410,26463,26466,26469,26472,26475,26478,26538,26541,26544,26547,26550,26553,26556,26559,26562,26565,26627,26630,26633,26636,26639,26642,26698,26701,26704,26707],{"className":26403},[2937],[507,26405,26407],{"className":26406,"style":2943},[2941,2942],[507,26408,2810],{"className":26409},[2947,9920],[507,26411,26413,26416],{"className":26412},[2570],[507,26414,3286],{"className":26415},[2570,2611],[507,26417,26419],{"className":26418},[2579],[507,26420,26422],{"className":26421},[2583],[507,26423,26425],{"className":26424},[2587],[507,26426,26428],{"className":26427,"style":5040},[2591],[507,26429,26430,26433],{"style":2906},[507,26431],{"className":26432,"style":2600},[2599],[507,26434,26436],{"className":26435},[2604,2605,2606,2607],[507,26437,26439,26442,26445,26448,26451,26454,26457,26460],{"className":26438},[2570,2607],[507,26440,3293],{"className":26441},[2570,2611,2607],[507,26443],{"className":26444,"style":25898},[2714,2607],[507,26446,4959],{"className":26447},[2570,2611,2607],[507,26449],{"className":26450,"style":25898},[2714,2607],[507,26452,3138],{"className":26453,"style":3153},[2570,2611,2607],[507,26455],{"className":26456,"style":25898},[2714,2607],[507,26458,4959],{"className":26459},[2570,2611,2607],[507,26461,3298],{"className":26462},[2570,2611,2607],[507,26464,2749],{"className":26465},[2570],[507,26467,3793],{"className":26468,"style":2776},[2570,2611],[507,26470,2756],{"className":26471},[2780],[507,26473,2819],{"className":26474},[2961],[507,26476],{"className":26477,"style":2965},[2714],[507,26479,26481,26484],{"className":26480},[2570],[507,26482,3286],{"className":26483},[2570,2611],[507,26485,26487],{"className":26486},[2579],[507,26488,26490],{"className":26489},[2583],[507,26491,26493],{"className":26492},[2587],[507,26494,26497],{"className":26495,"style":26496},[2591],"height:0.938em;",[507,26498,26499,26502],{"style":2906},[507,26500],{"className":26501,"style":2600},[2599],[507,26503,26505],{"className":26504},[2604,2605,2606,2607],[507,26506,26508,26511,26514,26517,26520,26523,26526,26529,26532,26535],{"className":26507},[2570,2607],[507,26509,3293],{"className":26510},[2570,2611,2607],[507,26512,580],{"className":26513},[2941,2607],[507,26515,4959],{"className":26516},[2570,2611,2607],[507,26518,2691],{"className":26519},[2719,2607],[507,26521,625],{"className":26522},[2570,2607],[507,26524,3649],{"className":26525},[2780,2607],[507,26527,3138],{"className":26528,"style":3153},[2570,2611,2607],[507,26530],{"className":26531,"style":25898},[2714,2607],[507,26533,4959],{"className":26534},[2570,2611,2607],[507,26536,3298],{"className":26537},[2570,2611,2607],[507,26539,2749],{"className":26540},[2570],[507,26542,3793],{"className":26543,"style":2776},[2570,2611],[507,26545,2756],{"className":26546},[2780],[507,26548,2819],{"className":26549},[2961],[507,26551],{"className":26552,"style":2965},[2714],[507,26554,25703],{"className":26555},[2937],[507,26557],{"className":26558,"style":2965},[2714],[507,26560,2819],{"className":26561},[2961],[507,26563],{"className":26564,"style":2965},[2714],[507,26566,26568,26571],{"className":26567},[2570],[507,26569,3286],{"className":26570},[2570,2611],[507,26572,26574],{"className":26573},[2579],[507,26575,26577],{"className":26576},[2583],[507,26578,26580],{"className":26579},[2587],[507,26581,26583],{"className":26582,"style":26496},[2591],[507,26584,26585,26588],{"style":2906},[507,26586],{"className":26587,"style":2600},[2599],[507,26589,26591],{"className":26590},[2604,2605,2606,2607],[507,26592,26594,26597,26600,26603,26606,26609,26612,26615,26618,26621,26624],{"className":26593},[2570,2607],[507,26595,2691],{"className":26596},[2570,2607],[507,26598,3293],{"className":26599},[2570,2611,2607],[507,26601,580],{"className":26602},[2941,2607],[507,26604,4959],{"className":26605},[2570,2611,2607],[507,26607,2691],{"className":26608},[2719,2607],[507,26610,625],{"className":26611},[2570,2607],[507,26613,3649],{"className":26614},[2780,2607],[507,26616,3138],{"className":26617,"style":3153},[2570,2611,2607],[507,26619],{"className":26620,"style":25898},[2714,2607],[507,26622,4959],{"className":26623},[2570,2611,2607],[507,26625,3298],{"className":26626},[2570,2611,2607],[507,26628,2749],{"className":26629},[2570],[507,26631,3793],{"className":26632,"style":2776},[2570,2611],[507,26634,2756],{"className":26635},[2780],[507,26637,2819],{"className":26638},[2961],[507,26640],{"className":26641,"style":2965},[2714],[507,26643,26645,26648],{"className":26644},[2570],[507,26646,3286],{"className":26647},[2570,2611],[507,26649,26651],{"className":26650},[2579],[507,26652,26654],{"className":26653},[2583],[507,26655,26657],{"className":26656},[2587],[507,26658,26660],{"className":26659,"style":5040},[2591],[507,26661,26662,26665],{"style":2906},[507,26663],{"className":26664,"style":2600},[2599],[507,26666,26668],{"className":26667},[2604,2605,2606,2607],[507,26669,26671,26674,26677,26680,26683,26686,26689,26692,26695],{"className":26670},[2570,2607],[507,26672,2691],{"className":26673},[2570,2607],[507,26675,3293],{"className":26676},[2570,2611,2607],[507,26678],{"className":26679,"style":25898},[2714,2607],[507,26681,4959],{"className":26682},[2570,2611,2607],[507,26684],{"className":26685,"style":25898},[2714,2607],[507,26687,3138],{"className":26688,"style":3153},[2570,2611,2607],[507,26690],{"className":26691,"style":25898},[2714,2607],[507,26693,4959],{"className":26694},[2570,2611,2607],[507,26696,3298],{"className":26697},[2570,2611,2607],[507,26699,2749],{"className":26700},[2570],[507,26702,3793],{"className":26703,"style":2776},[2570,2611],[507,26705,2756],{"className":26706},[2780],[507,26708,26710],{"className":26709,"style":2943},[2780,2942],[507,26711,2872],{"className":26712},[2947,9920],[507,26714],{"className":26715,"style":2965},[2714],[507,26717,2819],{"className":26718},[2961],[18,26720,26721,26722,26750],{},"that is, where all the time-evolutions are shifted backward by ",[507,26723,26725,26738],{"className":26724},[2523],[507,26726,26728],{"className":26727},[2527],[2529,26729,26730],{"xmlns":2531},[2533,26731,26732,26736],{},[2536,26733,26734],{},[2542,26735,4959],{},[2549,26737,4959],{"encoding":2551},[507,26739,26741],{"className":26740,"ariaHidden":2557},[2556],[507,26742,26744,26747],{"className":26743},[2561],[507,26745],{"className":26746,"style":5434},[2565],[507,26748,4959],{"className":26749},[2570,2611]," timesteps.\nThe reason it is indistinguishable is because the matrix elements",[507,26752,26754],{"className":26753},[2784],[507,26755,26757,26889],{"className":26756},[2523],[507,26758,26760],{"className":26759},[2527],[2529,26761,26762],{"xmlns":2531,"display":2793},[2533,26763,26764,26886],{},[2536,26765,26766,26782,26784,26786,26788,26790,26812,26814,26838,26840,26842,26844,26846,26848,26850,26852,26854,26880,26882,26884],{},[3168,26767,26768,26774],{},[4271,26769,26770,26772],{"accent":2557},[2542,26771,3138],{},[2689,26773,4277],{},[2536,26775,26776,26778,26780],{},[2542,26777,2372],{},[2689,26779,2819],{"separator":2557},[2542,26781,3626],{},[2689,26783,573],{},[2689,26785,4425],{"stretchy":2755},[2542,26787,3793],{},[2542,26789,2749],{"mathvariant":2748},[2539,26791,26792,26794],{},[2542,26793,3286],{},[2536,26795,26796,26798,26800,26802,26804,26806,26808,26810],{},[2542,26797,3293],{},[6167,26799,25688],{},[2542,26801,2372],{},[6167,26803,25688],{},[2542,26805,3138],{},[6167,26807,25688],{},[2542,26809,4959],{},[2542,26811,3298],{},[2542,26813,3138],{},[2539,26815,26816,26818],{},[2542,26817,3286],{},[2536,26819,26820,26822,26824,26826,26828,26830,26832,26834,26836],{},[2689,26821,2691],{},[2542,26823,3293],{},[6167,26825,25688],{},[2542,26827,3626],{},[6167,26829,25688],{},[2542,26831,3138],{},[6167,26833,25688],{},[2542,26835,4959],{},[2542,26837,3298],{},[2542,26839,2749],{"mathvariant":2748},[2542,26841,3793],{},[2689,26843,2756],{"stretchy":2755},[2689,26845,573],{},[2689,26847,4425],{"stretchy":2755},[2542,26849,3793],{},[2542,26851,2749],{"mathvariant":2748},[2542,26853,3138],{},[2539,26855,26856,26858],{},[2542,26857,3286],{},[2536,26859,26860,26862,26864,26866,26868,26870,26872,26874,26876,26878],{},[2542,26861,3293],{},[2689,26863,580],{"stretchy":2755},[2542,26865,2372],{},[2689,26867,2691],{},[2542,26869,3626],{},[2689,26871,3649],{"stretchy":2755},[2542,26873,3138],{},[6167,26875,25688],{},[2542,26877,4959],{},[2542,26879,3298],{},[2542,26881,2749],{"mathvariant":2748},[2542,26883,3793],{},[2689,26885,2756],{"stretchy":2755},[2549,26887,26888],{"encoding":2551},"\\tilde{H}_{j,k} = \\langle\\psi|e^{i\\,j\\,H\\,dt}He^{-i\\,k\\,H\\,dt}|\\psi\\rangle=\\langle\\psi|He^{i(j-k)H\\,dt}|\\psi\\rangle",[507,26890,26892,26984,27129],{"className":26891,"ariaHidden":2557},[2556],[507,26893,26895,26898,26975,26978,26981],{"className":26894},[2561],[507,26896],{"className":26897,"style":6755},[2565],[507,26899,26901,26932],{"className":26900},[2570],[507,26902,26904],{"className":26903},[2570,4294],[507,26905,26907],{"className":26906},[2583],[507,26908,26910],{"className":26909},[2587],[507,26911,26913,26921],{"className":26912,"style":4290},[2591],[507,26914,26915,26918],{"style":4306},[507,26916],{"className":26917,"style":4310},[2599],[507,26919,3138],{"className":26920,"style":3153},[2570,2611],[507,26922,26923,26926],{"style":4316},[507,26924],{"className":26925,"style":4310},[2599],[507,26927,26929],{"className":26928,"style":4324},[4323],[507,26930,4277],{"className":26931},[2570],[507,26933,26935],{"className":26934},[2579],[507,26936,26938,26967],{"className":26937},[2583,3200],[507,26939,26941,26964],{"className":26940},[2587],[507,26942,26944],{"className":26943,"style":11243},[2591],[507,26945,26946,26949],{"style":4510},[507,26947],{"className":26948,"style":2600},[2599],[507,26950,26952],{"className":26951},[2604,2605,2606,2607],[507,26953,26955,26958,26961],{"className":26954},[2570,2607],[507,26956,2372],{"className":26957,"style":6823},[2570,2611,2607],[507,26959,2819],{"className":26960},[2961,2607],[507,26962,3626],{"className":26963,"style":3692},[2570,2611,2607],[507,26965,3225],{"className":26966},[3224],[507,26968,26970],{"className":26969},[2587],[507,26971,26973],{"className":26972,"style":6833},[2591],[507,26974],{},[507,26976],{"className":26977,"style":2919},[2714],[507,26979,573],{"className":26980},[2923],[507,26982],{"className":26983,"style":2919},[2714],[507,26985,26987,26990,26993,26996,26999,27052,27055,27111,27114,27117,27120,27123,27126],{"className":26986},[2561],[507,26988],{"className":26989,"style":5012},[2565],[507,26991,4425],{"className":26992},[2941],[507,26994,3793],{"className":26995,"style":2776},[2570,2611],[507,26997,2749],{"className":26998},[2570],[507,27000,27002,27005],{"className":27001},[2570],[507,27003,3286],{"className":27004},[2570,2611],[507,27006,27008],{"className":27007},[2579],[507,27009,27011],{"className":27010},[2583],[507,27012,27014],{"className":27013},[2587],[507,27015,27017],{"className":27016,"style":5040},[2591],[507,27018,27019,27022],{"style":2906},[507,27020],{"className":27021,"style":2600},[2599],[507,27023,27025],{"className":27024},[2604,2605,2606,2607],[507,27026,27028,27031,27034,27037,27040,27043,27046,27049],{"className":27027},[2570,2607],[507,27029,3293],{"className":27030},[2570,2611,2607],[507,27032],{"className":27033,"style":25898},[2714,2607],[507,27035,2372],{"className":27036,"style":6823},[2570,2611,2607],[507,27038],{"className":27039,"style":25898},[2714,2607],[507,27041,3138],{"className":27042,"style":3153},[2570,2611,2607],[507,27044],{"className":27045,"style":25898},[2714,2607],[507,27047,4959],{"className":27048},[2570,2611,2607],[507,27050,3298],{"className":27051},[2570,2611,2607],[507,27053,3138],{"className":27054,"style":3153},[2570,2611],[507,27056,27058,27061],{"className":27057},[2570],[507,27059,3286],{"className":27060},[2570,2611],[507,27062,27064],{"className":27063},[2579],[507,27065,27067],{"className":27066},[2583],[507,27068,27070],{"className":27069},[2587],[507,27071,27073],{"className":27072,"style":5040},[2591],[507,27074,27075,27078],{"style":2906},[507,27076],{"className":27077,"style":2600},[2599],[507,27079,27081],{"className":27080},[2604,2605,2606,2607],[507,27082,27084,27087,27090,27093,27096,27099,27102,27105,27108],{"className":27083},[2570,2607],[507,27085,2691],{"className":27086},[2570,2607],[507,27088,3293],{"className":27089},[2570,2611,2607],[507,27091],{"className":27092,"style":25898},[2714,2607],[507,27094,3626],{"className":27095,"style":3692},[2570,2611,2607],[507,27097],{"className":27098,"style":25898},[2714,2607],[507,27100,3138],{"className":27101,"style":3153},[2570,2611,2607],[507,27103],{"className":27104,"style":25898},[2714,2607],[507,27106,4959],{"className":27107},[2570,2611,2607],[507,27109,3298],{"className":27110},[2570,2611,2607],[507,27112,2749],{"className":27113},[2570],[507,27115,3793],{"className":27116,"style":2776},[2570,2611],[507,27118,2756],{"className":27119},[2780],[507,27121],{"className":27122,"style":2919},[2714],[507,27124,573],{"className":27125},[2923],[507,27127],{"className":27128,"style":2919},[2714],[507,27130,27132,27136,27139,27142,27145,27148,27207,27210,27213],{"className":27131},[2561],[507,27133],{"className":27134,"style":27135},[2565],"height:1.188em;vertical-align:-0.25em;",[507,27137,4425],{"className":27138},[2941],[507,27140,3793],{"className":27141,"style":2776},[2570,2611],[507,27143,2749],{"className":27144},[2570],[507,27146,3138],{"className":27147,"style":3153},[2570,2611],[507,27149,27151,27154],{"className":27150},[2570],[507,27152,3286],{"className":27153},[2570,2611],[507,27155,27157],{"className":27156},[2579],[507,27158,27160],{"className":27159},[2583],[507,27161,27163],{"className":27162},[2587],[507,27164,27166],{"className":27165,"style":26496},[2591],[507,27167,27168,27171],{"style":2906},[507,27169],{"className":27170,"style":2600},[2599],[507,27172,27174],{"className":27173},[2604,2605,2606,2607],[507,27175,27177,27180,27183,27186,27189,27192,27195,27198,27201,27204],{"className":27176},[2570,2607],[507,27178,3293],{"className":27179},[2570,2611,2607],[507,27181,580],{"className":27182},[2941,2607],[507,27184,2372],{"className":27185,"style":6823},[2570,2611,2607],[507,27187,2691],{"className":27188},[2719,2607],[507,27190,3626],{"className":27191,"style":3692},[2570,2611,2607],[507,27193,3649],{"className":27194},[2780,2607],[507,27196,3138],{"className":27197,"style":3153},[2570,2611,2607],[507,27199],{"className":27200,"style":25898},[2714,2607],[507,27202,4959],{"className":27203},[2570,2611,2607],[507,27205,3298],{"className":27206},[2570,2611,2607],[507,27208,2749],{"className":27209},[2570],[507,27211,3793],{"className":27212,"style":2776},[2570,2611],[507,27214,2756],{"className":27215},[2780],[18,27217,27218,27219,27247,27248,53],{},"are invariant under overall shifts of the evolution time, since the time-evolutions commute with the Hamiltonian. For odd ",[507,27220,27222,27235],{"className":27221},[2523],[507,27223,27225],{"className":27224},[2527],[2529,27226,27227],{"xmlns":2531},[2533,27228,27229,27233],{},[2536,27230,27231],{},[2542,27232,2216],{},[2549,27234,2216],{"encoding":2551},[507,27236,27238],{"className":27237,"ariaHidden":2557},[2556],[507,27239,27241,27244],{"className":27240},[2561],[507,27242],{"className":27243,"style":2639},[2565],[507,27245,2216],{"className":27246,"style":2612},[2570,2611],", we can use the analysis for ",[507,27249,27251,27268],{"className":27250},[2523],[507,27252,27254],{"className":27253},[2527],[2529,27255,27256],{"xmlns":2531},[2533,27257,27258,27266],{},[2536,27259,27260,27262,27264],{},[2542,27261,2216],{},[2689,27263,2691],{},[2693,27265,625],{},[2549,27267,2697],{"encoding":2551},[507,27269,27271,27289],{"className":27270,"ariaHidden":2557},[2556],[507,27272,27274,27277,27280,27283,27286],{"className":27273},[2561],[507,27275],{"className":27276,"style":2707},[2565],[507,27278,2216],{"className":27279,"style":2612},[2570,2611],[507,27281],{"className":27282,"style":2715},[2714],[507,27284,2691],{"className":27285},[2719],[507,27287],{"className":27288,"style":2715},[2714],[507,27290,27292,27295],{"className":27291},[2561],[507,27293],{"className":27294,"style":2729},[2565],[507,27296,625],{"className":27297},[2570],[18,27299,27300,27301,24985],{},"We want to show that somewhere in this Krylov space, there is guaranteed to be a low-energy state. We do so by way of the following result, which is derived from Theorem 3.1 in ",[49,27302,27303],{"href":1086},"[3]",[18,27305,27306,27309,27310,27339,27340,27368],{},[154,27307,27308],{},"Claim 1:"," there exists a function ",[507,27311,27313,27326],{"className":27312},[2523],[507,27314,27316],{"className":27315},[2527],[2529,27317,27318],{"xmlns":2531},[2533,27319,27320,27324],{},[2536,27321,27322],{},[2542,27323,22278],{},[2549,27325,22278],{"encoding":2551},[507,27327,27329],{"className":27328,"ariaHidden":2557},[2556],[507,27330,27332,27335],{"className":27331},[2561],[507,27333],{"className":27334,"style":7035},[2565],[507,27336,22278],{"className":27337,"style":27338},[2570,2611],"margin-right:0.1076em;"," such that for energies ",[507,27341,27343,27356],{"className":27342},[2523],[507,27344,27346],{"className":27345},[2527],[2529,27347,27348],{"xmlns":2531},[2533,27349,27350,27354],{},[2536,27351,27352],{},[2542,27353,6182],{},[2549,27355,6182],{"encoding":2551},[507,27357,27359],{"className":27358,"ariaHidden":2557},[2556],[507,27360,27362,27365],{"className":27361},[2561],[507,27363],{"className":27364,"style":2566},[2565],[507,27366,6182],{"className":27367,"style":5892},[2570,2611]," in the spectral range of the Hamiltonian (that is, between the ground state energy and the maximum energy)...",[27370,27371,27372,27481,27789],"ol",{},[45,27373,27374],{},[507,27375,27377,27405],{"className":27376},[2523],[507,27378,27380],{"className":27379},[2527],[2529,27381,27382],{"xmlns":2531},[2533,27383,27384,27402],{},[2536,27385,27386,27388,27390,27396,27398,27400],{},[2542,27387,22278],{},[2689,27389,580],{"stretchy":2755},[3168,27391,27392,27394],{},[2542,27393,6182],{},[2693,27395,601],{},[2689,27397,3649],{"stretchy":2755},[2689,27399,573],{},[2693,27401,625],{},[2549,27403,27404],{"encoding":2551},"f(E_0)=1",[507,27406,27408,27472],{"className":27407,"ariaHidden":2557},[2556],[507,27409,27411,27414,27417,27420,27460,27463,27466,27469],{"className":27410},[2561],[507,27412],{"className":27413,"style":2769},[2565],[507,27415,22278],{"className":27416,"style":27338},[2570,2611],[507,27418,580],{"className":27419},[2941],[507,27421,27423,27426],{"className":27422},[2570],[507,27424,6182],{"className":27425,"style":5892},[2570,2611],[507,27427,27429],{"className":27428},[2579],[507,27430,27432,27452],{"className":27431},[2583,3200],[507,27433,27435,27449],{"className":27434},[2587],[507,27436,27438],{"className":27437,"style":14281},[2591],[507,27439,27440,27443],{"style":6014},[507,27441],{"className":27442,"style":2600},[2599],[507,27444,27446],{"className":27445},[2604,2605,2606,2607],[507,27447,601],{"className":27448},[2570,2607],[507,27450,3225],{"className":27451},[3224],[507,27453,27455],{"className":27454},[2587],[507,27456,27458],{"className":27457,"style":3232},[2591],[507,27459],{},[507,27461,3649],{"className":27462},[2780],[507,27464],{"className":27465,"style":2919},[2714],[507,27467,573],{"className":27468},[2923],[507,27470],{"className":27471,"style":2919},[2714],[507,27473,27475,27478],{"className":27474},[2561],[507,27476],{"className":27477,"style":2729},[2565],[507,27479,625],{"className":27480},[2570],[45,27482,27483,27643,27644,27672,27673,27718,27719,27788],{},[507,27484,27486,27536],{"className":27485},[2523],[507,27487,27489],{"className":27488},[2527],[2529,27490,27491],{"xmlns":2531},[2533,27492,27493,27533],{},[2536,27494,27495,27497,27499,27501,27503,27505,27507,27510,27512],{},[2542,27496,2749],{"mathvariant":2748},[2542,27498,22278],{},[2689,27500,580],{"stretchy":2755},[2542,27502,6182],{},[2689,27504,3649],{"stretchy":2755},[2542,27506,2749],{"mathvariant":2748},[2689,27508,27509],{},"≤",[2693,27511,584],{},[2539,27513,27514,27527],{},[2536,27515,27516,27518,27520,27522,27525],{},[2689,27517,580],{"fence":2557},[2693,27519,625],{},[2689,27521,2107],{},[2542,27523,27524],{},"δ",[2689,27526,3649],{"fence":2557},[2536,27528,27529,27531],{},[2689,27530,2691],{},[2542,27532,4959],{},[2549,27534,27535],{"encoding":2551},"|f(E)|\\le2\\left(1 + \\delta\\right)^{-d}",[507,27537,27539,27572],{"className":27538,"ariaHidden":2557},[2556],[507,27540,27542,27545,27548,27551,27554,27557,27560,27563,27566,27569],{"className":27541},[2561],[507,27543],{"className":27544,"style":2769},[2565],[507,27546,2749],{"className":27547},[2570],[507,27549,22278],{"className":27550,"style":27338},[2570,2611],[507,27552,580],{"className":27553},[2941],[507,27555,6182],{"className":27556,"style":5892},[2570,2611],[507,27558,3649],{"className":27559},[2780],[507,27561,2749],{"className":27562},[2570],[507,27564],{"className":27565,"style":2919},[2714],[507,27567,27509],{"className":27568},[2923],[507,27570],{"className":27571,"style":2919},[2714],[507,27573,27575,27579,27582,27585],{"className":27574},[2561],[507,27576],{"className":27577,"style":27578},[2565],"height:1.239em;vertical-align:-0.25em;",[507,27580,584],{"className":27581},[2570],[507,27583],{"className":27584,"style":2965},[2714],[507,27586,27588,27613],{"className":27587},[2937],[507,27589,27591,27594,27597,27600,27603,27606,27610],{"className":27590},[2937],[507,27592,580],{"className":27593,"style":2943},[2941,2942],[507,27595,625],{"className":27596},[2570],[507,27598],{"className":27599,"style":2715},[2714],[507,27601,2107],{"className":27602},[2719],[507,27604],{"className":27605,"style":2715},[2714],[507,27607,27524],{"className":27608,"style":27609},[2570,2611],"margin-right:0.0379em;",[507,27611,3649],{"className":27612,"style":2943},[2780,2942],[507,27614,27616],{"className":27615},[2579],[507,27617,27619],{"className":27618},[2583],[507,27620,27622],{"className":27621},[2587],[507,27623,27626],{"className":27624,"style":27625},[2591],"height:0.989em;",[507,27627,27628,27631],{"style":7845},[507,27629],{"className":27630,"style":2600},[2599],[507,27632,27634],{"className":27633},[2604,2605,2606,2607],[507,27635,27637,27640],{"className":27636},[2570,2607],[507,27638,2691],{"className":27639},[2570,2607],[507,27641,4959],{"className":27642},[2570,2611,2607]," for all values of ",[507,27645,27647,27660],{"className":27646},[2523],[507,27648,27650],{"className":27649},[2527],[2529,27651,27652],{"xmlns":2531},[2533,27653,27654,27658],{},[2536,27655,27656],{},[2542,27657,6182],{},[2549,27659,6182],{"encoding":2551},[507,27661,27663],{"className":27662,"ariaHidden":2557},[2556],[507,27664,27666,27669],{"className":27665},[2561],[507,27667],{"className":27668,"style":2566},[2565],[507,27670,6182],{"className":27671,"style":5892},[2570,2611]," that lie ",[507,27674,27676,27693],{"className":27675},[2523],[507,27677,27679],{"className":27678},[2527],[2529,27680,27681],{"xmlns":2531},[2533,27682,27683,27690],{},[2536,27684,27685,27688],{},[2689,27686,27687],{},"≥",[2542,27689,27524],{},[2549,27691,27692],{"encoding":2551},"\\ge\\delta",[507,27694,27696,27709],{"className":27695,"ariaHidden":2557},[2556],[507,27697,27699,27703,27706],{"className":27698},[2561],[507,27700],{"className":27701,"style":27702},[2565],"height:0.7719em;vertical-align:-0.136em;",[507,27704,27687],{"className":27705},[2923],[507,27707],{"className":27708,"style":2919},[2714],[507,27710,27712,27715],{"className":27711},[2561],[507,27713],{"className":27714,"style":5434},[2565],[507,27716,27524],{"className":27717,"style":27609},[2570,2611]," away from ",[507,27720,27722,27739],{"className":27721},[2523],[507,27723,27725],{"className":27724},[2527],[2529,27726,27727],{"xmlns":2531},[2533,27728,27729,27737],{},[2536,27730,27731],{},[3168,27732,27733,27735],{},[2542,27734,6182],{},[2693,27736,601],{},[2549,27738,14253],{"encoding":2551},[507,27740,27742],{"className":27741,"ariaHidden":2557},[2556],[507,27743,27745,27748],{"className":27744},[2561],[507,27746],{"className":27747,"style":3187},[2565],[507,27749,27751,27754],{"className":27750},[2570],[507,27752,6182],{"className":27753,"style":5892},[2570,2611],[507,27755,27757],{"className":27756},[2579],[507,27758,27760,27780],{"className":27759},[2583,3200],[507,27761,27763,27777],{"className":27762},[2587],[507,27764,27766],{"className":27765,"style":14281},[2591],[507,27767,27768,27771],{"style":6014},[507,27769],{"className":27770,"style":2600},[2599],[507,27772,27774],{"className":27773},[2604,2605,2606,2607],[507,27775,601],{"className":27776},[2570,2607],[507,27778,3225],{"className":27779},[3224],[507,27781,27783],{"className":27782},[2587],[507,27784,27786],{"className":27785,"style":3232},[2591],[507,27787],{},", that is, it is exponentially suppressed",[45,27790,27791,27835,27836,27923,27924],{},[507,27792,27794,27814],{"className":27793},[2523],[507,27795,27797],{"className":27796},[2527],[2529,27798,27799],{"xmlns":2531},[2533,27800,27801,27811],{},[2536,27802,27803,27805,27807,27809],{},[2542,27804,22278],{},[2689,27806,580],{"stretchy":2755},[2542,27808,6182],{},[2689,27810,3649],{"stretchy":2755},[2549,27812,27813],{"encoding":2551},"f(E)",[507,27815,27817],{"className":27816,"ariaHidden":2557},[2556],[507,27818,27820,27823,27826,27829,27832],{"className":27819},[2561],[507,27821],{"className":27822,"style":2769},[2565],[507,27824,22278],{"className":27825,"style":27338},[2570,2611],[507,27827,580],{"className":27828},[2941],[507,27830,6182],{"className":27831,"style":5892},[2570,2611],[507,27833,3649],{"className":27834},[2780]," is a linear combination of ",[507,27837,27839,27869],{"className":27838},[2523],[507,27840,27842],{"className":27841},[2527],[2529,27843,27844],{"xmlns":2531},[2533,27845,27846,27866],{},[2536,27847,27848],{},[2539,27849,27850,27852],{},[2542,27851,3286],{},[2536,27853,27854,27856,27858,27860,27862,27864],{},[2542,27855,3293],{},[2542,27857,2372],{},[2542,27859,6182],{},[6167,27861,25688],{},[2542,27863,4959],{},[2542,27865,3298],{},[2549,27867,27868],{"encoding":2551},"e^{ijE\\,dt}",[507,27870,27872],{"className":27871,"ariaHidden":2557},[2556],[507,27873,27875,27878],{"className":27874},[2561],[507,27876],{"className":27877,"style":3662},[2565],[507,27879,27881,27884],{"className":27880},[2570],[507,27882,3286],{"className":27883},[2570,2611],[507,27885,27887],{"className":27886},[2579],[507,27888,27890],{"className":27889},[2583],[507,27891,27893],{"className":27892},[2587],[507,27894,27896],{"className":27895,"style":3662},[2591],[507,27897,27898,27901],{"style":2595},[507,27899],{"className":27900,"style":2600},[2599],[507,27902,27904],{"className":27903},[2604,2605,2606,2607],[507,27905,27907,27911,27914,27917,27920],{"className":27906},[2570,2607],[507,27908,27910],{"className":27909,"style":6823},[2570,2611,2607],"ij",[507,27912,6182],{"className":27913,"style":5892},[2570,2611,2607],[507,27915],{"className":27916,"style":25898},[2714,2607],[507,27918,4959],{"className":27919},[2570,2611,2607],[507,27921,3298],{"className":27922},[2570,2611,2607]," for ",[507,27925,27927,27977],{"className":27926},[2523],[507,27928,27930],{"className":27929},[2527],[2529,27931,27932],{"xmlns":2531},[2533,27933,27934,27974],{},[2536,27935,27936,27938,27940,27942,27944,27946,27948,27950,27952,27954,27956,27958,27960,27962,27964,27966,27968,27970,27972],{},[2542,27937,2372],{},[2689,27939,573],{},[2689,27941,2691],{},[2542,27943,4959],{},[2689,27945,2819],{"separator":2557},[2689,27947,2691],{},[2542,27949,4959],{},[2689,27951,2107],{},[2693,27953,625],{},[2689,27955,2819],{"separator":2557},[2542,27957,53],{"mathvariant":2748},[2542,27959,53],{"mathvariant":2748},[2542,27961,53],{"mathvariant":2748},[2689,27963,2819],{"separator":2557},[2542,27965,4959],{},[2689,27967,2691],{},[2693,27969,625],{},[2689,27971,2819],{"separator":2557},[2542,27973,4959],{},[2549,27975,27976],{"encoding":2551},"j=-d,-d+1,...,d-1,d",[507,27978,27980,27999,28032,28068],{"className":27979,"ariaHidden":2557},[2556],[507,27981,27983,27987,27990,27993,27996],{"className":27982},[2561],[507,27984],{"className":27985,"style":27986},[2565],"height:0.854em;vertical-align:-0.1944em;",[507,27988,2372],{"className":27989,"style":6823},[2570,2611],[507,27991],{"className":27992,"style":2919},[2714],[507,27994,573],{"className":27995},[2923],[507,27997],{"className":27998,"style":2919},[2714],[507,28000,28002,28005,28008,28011,28014,28017,28020,28023,28026,28029],{"className":28001},[2561],[507,28003],{"className":28004,"style":7035},[2565],[507,28006,2691],{"className":28007},[2570],[507,28009,4959],{"className":28010},[2570,2611],[507,28012,2819],{"className":28013},[2961],[507,28015],{"className":28016,"style":2965},[2714],[507,28018,2691],{"className":28019},[2570],[507,28021,4959],{"className":28022},[2570,2611],[507,28024],{"className":28025,"style":2715},[2714],[507,28027,2107],{"className":28028},[2719],[507,28030],{"className":28031,"style":2715},[2714],[507,28033,28035,28038,28041,28044,28047,28050,28053,28056,28059,28062,28065],{"className":28034},[2561],[507,28036],{"className":28037,"style":7035},[2565],[507,28039,625],{"className":28040},[2570],[507,28042,2819],{"className":28043},[2961],[507,28045],{"className":28046,"style":2965},[2714],[507,28048,3032],{"className":28049},[2570],[507,28051,2819],{"className":28052},[2961],[507,28054],{"className":28055,"style":2965},[2714],[507,28057,4959],{"className":28058},[2570,2611],[507,28060],{"className":28061,"style":2715},[2714],[507,28063,2691],{"className":28064},[2719],[507,28066],{"className":28067,"style":2715},[2714],[507,28069,28071,28074,28077,28080,28083],{"className":28070},[2561],[507,28072],{"className":28073,"style":7035},[2565],[507,28075,625],{"className":28076},[2570],[507,28078,2819],{"className":28079},[2961],[507,28081],{"className":28082,"style":2965},[2714],[507,28084,4959],{"className":28085},[2570,2611],[18,28087,28088,28089,28148,28149,28188],{},"We give a proof below, but that can be safely skipped unless one wants to understand the full, rigorous argument. For now we focus on the implications of the above claim. By property 3 above, we can see that the shifted Krylov space above contains the state ",[507,28090,28092,28118],{"className":28091},[2523],[507,28093,28095],{"className":28094},[2527],[2529,28096,28097],{"xmlns":2531},[2533,28098,28099,28115],{},[2536,28100,28101,28103,28105,28107,28109,28111,28113],{},[2542,28102,22278],{},[2689,28104,580],{"stretchy":2755},[2542,28106,3138],{},[2689,28108,3649],{"stretchy":2755},[2542,28110,2749],{"mathvariant":2748},[2542,28112,3793],{},[2689,28114,2756],{"stretchy":2755},[2549,28116,28117],{"encoding":2551},"f(H)|\\psi\\rangle",[507,28119,28121],{"className":28120,"ariaHidden":2557},[2556],[507,28122,28124,28127,28130,28133,28136,28139,28142,28145],{"className":28123},[2561],[507,28125],{"className":28126,"style":2769},[2565],[507,28128,22278],{"className":28129,"style":27338},[2570,2611],[507,28131,580],{"className":28132},[2941],[507,28134,3138],{"className":28135,"style":3153},[2570,2611],[507,28137,3649],{"className":28138},[2780],[507,28140,2749],{"className":28141},[2570],[507,28143,3793],{"className":28144,"style":2776},[2570,2611],[507,28146,2756],{"className":28147},[2780],". This is our low-energy state. To see why, write ",[507,28150,28152,28170],{"className":28151},[2523],[507,28153,28155],{"className":28154},[2527],[2529,28156,28157],{"xmlns":2531},[2533,28158,28159,28167],{},[2536,28160,28161,28163,28165],{},[2542,28162,2749],{"mathvariant":2748},[2542,28164,3793],{},[2689,28166,2756],{"stretchy":2755},[2549,28168,28169],{"encoding":2551},"|\\psi\\rangle",[507,28171,28173],{"className":28172,"ariaHidden":2557},[2556],[507,28174,28176,28179,28182,28185],{"className":28175},[2561],[507,28177],{"className":28178,"style":2769},[2565],[507,28180,2749],{"className":28181},[2570],[507,28183,3793],{"className":28184,"style":2776},[2570,2611],[507,28186,2756],{"className":28187},[2780]," in the energy eigenbasis:",[507,28190,28192],{"className":28191},[2784],[507,28193,28195,28249],{"className":28194},[2523],[507,28196,28198],{"className":28197},[2527],[2529,28199,28200],{"xmlns":2531,"display":2793},[2533,28201,28202,28246],{},[2536,28203,28204,28206,28208,28210,28212,28227,28234,28236,28242,28244],{},[2542,28205,2749],{"mathvariant":2748},[2542,28207,3793],{},[2689,28209,2756],{"stretchy":2755},[2689,28211,573],{},[28213,28214,28215,28217,28225],"munderover",{},[2689,28216,7527],{},[2536,28218,28219,28221,28223],{},[2542,28220,3626],{},[2689,28222,573],{},[2693,28224,601],{},[2542,28226,7681],{},[3168,28228,28229,28232],{},[2542,28230,28231],{},"γ",[2542,28233,3626],{},[2542,28235,2749],{"mathvariant":2748},[3168,28237,28238,28240],{},[2542,28239,6182],{},[2542,28241,3626],{},[2689,28243,2756],{"stretchy":2755},[2689,28245,2819],{"separator":2557},[2549,28247,28248],{"encoding":2551},"|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_k|E_k\\rangle,",[507,28250,28252,28276],{"className":28251,"ariaHidden":2557},[2556],[507,28253,28255,28258,28261,28264,28267,28270,28273],{"className":28254},[2561],[507,28256],{"className":28257,"style":2769},[2565],[507,28259,2749],{"className":28260},[2570],[507,28262,3793],{"className":28263,"style":2776},[2570,2611],[507,28265,2756],{"className":28266},[2780],[507,28268],{"className":28269,"style":2919},[2714],[507,28271,573],{"className":28272},[2923],[507,28274],{"className":28275,"style":2919},[2714],[507,28277,28279,28283,28358,28361,28403,28406,28446,28449],{"className":28278},[2561],[507,28280],{"className":28281,"style":28282},[2565],"height:3.1304em;vertical-align:-1.3021em;",[507,28284,28287],{"className":28285},[7570,28286],"op-limits",[507,28288,28290,28349],{"className":28289},[2583,3200],[507,28291,28293,28346],{"className":28292},[2587],[507,28294,28297,28319,28331],{"className":28295,"style":28296},[2591],"height:1.8283em;",[507,28298,28300,28304],{"style":28299},"top:-1.8479em;margin-left:0em;",[507,28301],{"className":28302,"style":28303},[2599],"height:3.05em;",[507,28305,28307],{"className":28306},[2604,2605,2606,2607],[507,28308,28310,28313,28316],{"className":28309},[2570,2607],[507,28311,3626],{"className":28312,"style":3692},[2570,2611,2607],[507,28314,573],{"className":28315},[2923,2607],[507,28317,601],{"className":28318},[2570,2607],[507,28320,28322,28325],{"style":28321},"top:-3.05em;",[507,28323],{"className":28324,"style":28303},[2599],[507,28326,28327],{},[507,28328,7527],{"className":28329},[7570,7574,28330],"large-op",[507,28332,28334,28337],{"style":28333},"top:-4.3em;margin-left:0em;",[507,28335],{"className":28336,"style":28303},[2599],[507,28338,28340],{"className":28339},[2604,2605,2606,2607],[507,28341,28343],{"className":28342},[2570,2607],[507,28344,7681],{"className":28345,"style":3314},[2570,2611,2607],[507,28347,3225],{"className":28348},[3224],[507,28350,28352],{"className":28351},[2587],[507,28353,28356],{"className":28354,"style":28355},[2591],"height:1.3021em;",[507,28357],{},[507,28359],{"className":28360,"style":2965},[2714],[507,28362,28364,28368],{"className":28363},[2570],[507,28365,28231],{"className":28366,"style":28367},[2570,2611],"margin-right:0.0556em;",[507,28369,28371],{"className":28370},[2579],[507,28372,28374,28395],{"className":28373},[2583,3200],[507,28375,28377,28392],{"className":28376},[2587],[507,28378,28380],{"className":28379,"style":11243},[2591],[507,28381,28383,28386],{"style":28382},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[507,28384],{"className":28385,"style":2600},[2599],[507,28387,28389],{"className":28388},[2604,2605,2606,2607],[507,28390,3626],{"className":28391,"style":3692},[2570,2611,2607],[507,28393,3225],{"className":28394},[3224],[507,28396,28398],{"className":28397},[2587],[507,28399,28401],{"className":28400,"style":3232},[2591],[507,28402],{},[507,28404,2749],{"className":28405},[2570],[507,28407,28409,28412],{"className":28408},[2570],[507,28410,6182],{"className":28411,"style":5892},[2570,2611],[507,28413,28415],{"className":28414},[2579],[507,28416,28418,28438],{"className":28417},[2583,3200],[507,28419,28421,28435],{"className":28420},[2587],[507,28422,28424],{"className":28423,"style":11243},[2591],[507,28425,28426,28429],{"style":6014},[507,28427],{"className":28428,"style":2600},[2599],[507,28430,28432],{"className":28431},[2604,2605,2606,2607],[507,28433,3626],{"className":28434,"style":3692},[2570,2611,2607],[507,28436,3225],{"className":28437},[3224],[507,28439,28441],{"className":28440},[2587],[507,28442,28444],{"className":28443,"style":3232},[2591],[507,28445],{},[507,28447,2756],{"className":28448},[2780],[507,28450,2819],{"className":28451},[2961],[18,28453,28454,28455,28535,28536,28607,28608,28646,28647,28705],{},"where ",[507,28456,28458,28480],{"className":28457},[2523],[507,28459,28461],{"className":28460},[2527],[2529,28462,28463],{"xmlns":2531},[2533,28464,28465,28477],{},[2536,28466,28467,28469,28475],{},[2542,28468,2749],{"mathvariant":2748},[3168,28470,28471,28473],{},[2542,28472,6182],{},[2542,28474,3626],{},[2689,28476,2756],{"stretchy":2755},[2549,28478,28479],{"encoding":2551},"|E_k\\rangle",[507,28481,28483],{"className":28482,"ariaHidden":2557},[2556],[507,28484,28486,28489,28492,28532],{"className":28485},[2561],[507,28487],{"className":28488,"style":2769},[2565],[507,28490,2749],{"className":28491},[2570],[507,28493,28495,28498],{"className":28494},[2570],[507,28496,6182],{"className":28497,"style":5892},[2570,2611],[507,28499,28501],{"className":28500},[2579],[507,28502,28504,28524],{"className":28503},[2583,3200],[507,28505,28507,28521],{"className":28506},[2587],[507,28508,28510],{"className":28509,"style":11243},[2591],[507,28511,28512,28515],{"style":6014},[507,28513],{"className":28514,"style":2600},[2599],[507,28516,28518],{"className":28517},[2604,2605,2606,2607],[507,28519,3626],{"className":28520,"style":3692},[2570,2611,2607],[507,28522,3225],{"className":28523},[3224],[507,28525,28527],{"className":28526},[2587],[507,28528,28530],{"className":28529,"style":3232},[2591],[507,28531],{},[507,28533,2756],{"className":28534},[2780]," is the kth energy eigenstate and ",[507,28537,28539,28557],{"className":28538},[2523],[507,28540,28542],{"className":28541},[2527],[2529,28543,28544],{"xmlns":2531},[2533,28545,28546,28554],{},[2536,28547,28548],{},[3168,28549,28550,28552],{},[2542,28551,28231],{},[2542,28553,3626],{},[2549,28555,28556],{"encoding":2551},"\\gamma_k",[507,28558,28560],{"className":28559,"ariaHidden":2557},[2556],[507,28561,28563,28567],{"className":28562},[2561],[507,28564],{"className":28565,"style":28566},[2565],"height:0.625em;vertical-align:-0.1944em;",[507,28568,28570,28573],{"className":28569},[2570],[507,28571,28231],{"className":28572,"style":28367},[2570,2611],[507,28574,28576],{"className":28575},[2579],[507,28577,28579,28599],{"className":28578},[2583,3200],[507,28580,28582,28596],{"className":28581},[2587],[507,28583,28585],{"className":28584,"style":11243},[2591],[507,28586,28587,28590],{"style":28382},[507,28588],{"className":28589,"style":2600},[2599],[507,28591,28593],{"className":28592},[2604,2605,2606,2607],[507,28594,3626],{"className":28595,"style":3692},[2570,2611,2607],[507,28597,3225],{"className":28598},[3224],[507,28600,28602],{"className":28601},[2587],[507,28603,28605],{"className":28604,"style":3232},[2591],[507,28606],{}," is its amplitude in the initial state ",[507,28609,28611,28628],{"className":28610},[2523],[507,28612,28614],{"className":28613},[2527],[2529,28615,28616],{"xmlns":2531},[2533,28617,28618,28626],{},[2536,28619,28620,28622,28624],{},[2542,28621,2749],{"mathvariant":2748},[2542,28623,3793],{},[2689,28625,2756],{"stretchy":2755},[2549,28627,28169],{"encoding":2551},[507,28629,28631],{"className":28630,"ariaHidden":2557},[2556],[507,28632,28634,28637,28640,28643],{"className":28633},[2561],[507,28635],{"className":28636,"style":2769},[2565],[507,28638,2749],{"className":28639},[2570],[507,28641,3793],{"className":28642,"style":2776},[2570,2611],[507,28644,2756],{"className":28645},[2780],". Expressed in terms of this, ",[507,28648,28650,28675],{"className":28649},[2523],[507,28651,28653],{"className":28652},[2527],[2529,28654,28655],{"xmlns":2531},[2533,28656,28657,28673],{},[2536,28658,28659,28661,28663,28665,28667,28669,28671],{},[2542,28660,22278],{},[2689,28662,580],{"stretchy":2755},[2542,28664,3138],{},[2689,28666,3649],{"stretchy":2755},[2542,28668,2749],{"mathvariant":2748},[2542,28670,3793],{},[2689,28672,2756],{"stretchy":2755},[2549,28674,28117],{"encoding":2551},[507,28676,28678],{"className":28677,"ariaHidden":2557},[2556],[507,28679,28681,28684,28687,28690,28693,28696,28699,28702],{"className":28680},[2561],[507,28682],{"className":28683,"style":2769},[2565],[507,28685,22278],{"className":28686,"style":27338},[2570,2611],[507,28688,580],{"className":28689},[2941],[507,28691,3138],{"className":28692,"style":3153},[2570,2611],[507,28694,3649],{"className":28695},[2780],[507,28697,2749],{"className":28698},[2570],[507,28700,3793],{"className":28701,"style":2776},[2570,2611],[507,28703,2756],{"className":28704},[2780]," is given by",[507,28707,28709],{"className":28708},[2784],[507,28710,28712,28784],{"className":28711},[2523],[507,28713,28715],{"className":28714},[2527],[2529,28716,28717],{"xmlns":2531,"display":2793},[2533,28718,28719,28781],{},[2536,28720,28721,28723,28725,28727,28729,28731,28733,28735,28737,28751,28757,28759,28761,28767,28769,28771,28777,28779],{},[2542,28722,22278],{},[2689,28724,580],{"stretchy":2755},[2542,28726,3138],{},[2689,28728,3649],{"stretchy":2755},[2542,28730,2749],{"mathvariant":2748},[2542,28732,3793],{},[2689,28734,2756],{"stretchy":2755},[2689,28736,573],{},[28213,28738,28739,28741,28749],{},[2689,28740,7527],{},[2536,28742,28743,28745,28747],{},[2542,28744,3626],{},[2689,28746,573],{},[2693,28748,601],{},[2542,28750,7681],{},[3168,28752,28753,28755],{},[2542,28754,28231],{},[2542,28756,3626],{},[2542,28758,22278],{},[2689,28760,580],{"stretchy":2755},[3168,28762,28763,28765],{},[2542,28764,6182],{},[2542,28766,3626],{},[2689,28768,3649],{"stretchy":2755},[2542,28770,2749],{"mathvariant":2748},[3168,28772,28773,28775],{},[2542,28774,6182],{},[2542,28776,3626],{},[2689,28778,2756],{"stretchy":2755},[2689,28780,2819],{"separator":2557},[2549,28782,28783],{"encoding":2551},"f(H)|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_kf(E_k)|E_k\\rangle,",[507,28785,28787,28823],{"className":28786,"ariaHidden":2557},[2556],[507,28788,28790,28793,28796,28799,28802,28805,28808,28811,28814,28817,28820],{"className":28789},[2561],[507,28791],{"className":28792,"style":2769},[2565],[507,28794,22278],{"className":28795,"style":27338},[2570,2611],[507,28797,580],{"className":28798},[2941],[507,28800,3138],{"className":28801,"style":3153},[2570,2611],[507,28803,3649],{"className":28804},[2780],[507,28806,2749],{"className":28807},[2570],[507,28809,3793],{"className":28810,"style":2776},[2570,2611],[507,28812,2756],{"className":28813},[2780],[507,28815],{"className":28816,"style":2919},[2714],[507,28818,573],{"className":28819},[2923],[507,28821],{"className":28822,"style":2919},[2714],[507,28824,28826,28829,28896,28899,28939,28942,28945,28985,28988,28991,29031,29034],{"className":28825},[2561],[507,28827],{"className":28828,"style":28282},[2565],[507,28830,28832],{"className":28831},[7570,28286],[507,28833,28835,28888],{"className":28834},[2583,3200],[507,28836,28838,28885],{"className":28837},[2587],[507,28839,28841,28861,28871],{"className":28840,"style":28296},[2591],[507,28842,28843,28846],{"style":28299},[507,28844],{"className":28845,"style":28303},[2599],[507,28847,28849],{"className":28848},[2604,2605,2606,2607],[507,28850,28852,28855,28858],{"className":28851},[2570,2607],[507,28853,3626],{"className":28854,"style":3692},[2570,2611,2607],[507,28856,573],{"className":28857},[2923,2607],[507,28859,601],{"className":28860},[2570,2607],[507,28862,28863,28866],{"style":28321},[507,28864],{"className":28865,"style":28303},[2599],[507,28867,28868],{},[507,28869,7527],{"className":28870},[7570,7574,28330],[507,28872,28873,28876],{"style":28333},[507,28874],{"className":28875,"style":28303},[2599],[507,28877,28879],{"className":28878},[2604,2605,2606,2607],[507,28880,28882],{"className":28881},[2570,2607],[507,28883,7681],{"className":28884,"style":3314},[2570,2611,2607],[507,28886,3225],{"className":28887},[3224],[507,28889,28891],{"className":28890},[2587],[507,28892,28894],{"className":28893,"style":28355},[2591],[507,28895],{},[507,28897],{"className":28898,"style":2965},[2714],[507,28900,28902,28905],{"className":28901},[2570],[507,28903,28231],{"className":28904,"style":28367},[2570,2611],[507,28906,28908],{"className":28907},[2579],[507,28909,28911,28931],{"className":28910},[2583,3200],[507,28912,28914,28928],{"className":28913},[2587],[507,28915,28917],{"className":28916,"style":11243},[2591],[507,28918,28919,28922],{"style":28382},[507,28920],{"className":28921,"style":2600},[2599],[507,28923,28925],{"className":28924},[2604,2605,2606,2607],[507,28926,3626],{"className":28927,"style":3692},[2570,2611,2607],[507,28929,3225],{"className":28930},[3224],[507,28932,28934],{"className":28933},[2587],[507,28935,28937],{"className":28936,"style":3232},[2591],[507,28938],{},[507,28940,22278],{"className":28941,"style":27338},[2570,2611],[507,28943,580],{"className":28944},[2941],[507,28946,28948,28951],{"className":28947},[2570],[507,28949,6182],{"className":28950,"style":5892},[2570,2611],[507,28952,28954],{"className":28953},[2579],[507,28955,28957,28977],{"className":28956},[2583,3200],[507,28958,28960,28974],{"className":28959},[2587],[507,28961,28963],{"className":28962,"style":11243},[2591],[507,28964,28965,28968],{"style":6014},[507,28966],{"className":28967,"style":2600},[2599],[507,28969,28971],{"className":28970},[2604,2605,2606,2607],[507,28972,3626],{"className":28973,"style":3692},[2570,2611,2607],[507,28975,3225],{"className":28976},[3224],[507,28978,28980],{"className":28979},[2587],[507,28981,28983],{"className":28982,"style":3232},[2591],[507,28984],{},[507,28986,3649],{"className":28987},[2780],[507,28989,2749],{"className":28990},[2570],[507,28992,28994,28997],{"className":28993},[2570],[507,28995,6182],{"className":28996,"style":5892},[2570,2611],[507,28998,29000],{"className":28999},[2579],[507,29001,29003,29023],{"className":29002},[2583,3200],[507,29004,29006,29020],{"className":29005},[2587],[507,29007,29009],{"className":29008,"style":11243},[2591],[507,29010,29011,29014],{"style":6014},[507,29012],{"className":29013,"style":2600},[2599],[507,29015,29017],{"className":29016},[2604,2605,2606,2607],[507,29018,3626],{"className":29019,"style":3692},[2570,2611,2607],[507,29021,3225],{"className":29022},[3224],[507,29024,29026],{"className":29025},[2587],[507,29027,29029],{"className":29028,"style":3232},[2591],[507,29030],{},[507,29032,2756],{"className":29033},[2780],[507,29035,2819],{"className":29036},[2961],[18,29038,29039,29040,29068,29069,29139,29140,29219],{},"using the fact that we can replace ",[507,29041,29043,29056],{"className":29042},[2523],[507,29044,29046],{"className":29045},[2527],[2529,29047,29048],{"xmlns":2531},[2533,29049,29050,29054],{},[2536,29051,29052],{},[2542,29053,3138],{},[2549,29055,3138],{"encoding":2551},[507,29057,29059],{"className":29058,"ariaHidden":2557},[2556],[507,29060,29062,29065],{"className":29061},[2561],[507,29063],{"className":29064,"style":2566},[2565],[507,29066,3138],{"className":29067,"style":3153},[2570,2611]," by ",[507,29070,29072,29090],{"className":29071},[2523],[507,29073,29075],{"className":29074},[2527],[2529,29076,29077],{"xmlns":2531},[2533,29078,29079,29087],{},[2536,29080,29081],{},[3168,29082,29083,29085],{},[2542,29084,6182],{},[2542,29086,3626],{},[2549,29088,29089],{"encoding":2551},"E_k",[507,29091,29093],{"className":29092,"ariaHidden":2557},[2556],[507,29094,29096,29099],{"className":29095},[2561],[507,29097],{"className":29098,"style":3187},[2565],[507,29100,29102,29105],{"className":29101},[2570],[507,29103,6182],{"className":29104,"style":5892},[2570,2611],[507,29106,29108],{"className":29107},[2579],[507,29109,29111,29131],{"className":29110},[2583,3200],[507,29112,29114,29128],{"className":29113},[2587],[507,29115,29117],{"className":29116,"style":11243},[2591],[507,29118,29119,29122],{"style":6014},[507,29120],{"className":29121,"style":2600},[2599],[507,29123,29125],{"className":29124},[2604,2605,2606,2607],[507,29126,3626],{"className":29127,"style":3692},[2570,2611,2607],[507,29129,3225],{"className":29130},[3224],[507,29132,29134],{"className":29133},[2587],[507,29135,29137],{"className":29136,"style":3232},[2591],[507,29138],{}," when it acts on the eigenstate ",[507,29141,29143,29164],{"className":29142},[2523],[507,29144,29146],{"className":29145},[2527],[2529,29147,29148],{"xmlns":2531},[2533,29149,29150,29162],{},[2536,29151,29152,29154,29160],{},[2542,29153,2749],{"mathvariant":2748},[3168,29155,29156,29158],{},[2542,29157,6182],{},[2542,29159,3626],{},[2689,29161,2756],{"stretchy":2755},[2549,29163,28479],{"encoding":2551},[507,29165,29167],{"className":29166,"ariaHidden":2557},[2556],[507,29168,29170,29173,29176,29216],{"className":29169},[2561],[507,29171],{"className":29172,"style":2769},[2565],[507,29174,2749],{"className":29175},[2570],[507,29177,29179,29182],{"className":29178},[2570],[507,29180,6182],{"className":29181,"style":5892},[2570,2611],[507,29183,29185],{"className":29184},[2579],[507,29186,29188,29208],{"className":29187},[2583,3200],[507,29189,29191,29205],{"className":29190},[2587],[507,29192,29194],{"className":29193,"style":11243},[2591],[507,29195,29196,29199],{"style":6014},[507,29197],{"className":29198,"style":2600},[2599],[507,29200,29202],{"className":29201},[2604,2605,2606,2607],[507,29203,3626],{"className":29204,"style":3692},[2570,2611,2607],[507,29206,3225],{"className":29207},[3224],[507,29209,29211],{"className":29210},[2587],[507,29212,29214],{"className":29213,"style":3232},[2591],[507,29215],{},[507,29217,2756],{"className":29218},[2780],". The energy error of this state is therefore",[507,29221,29223],{"className":29222},[2784],[507,29224,29226,29315],{"className":29225},[2523],[507,29227,29229],{"className":29228},[2527],[2529,29230,29231],{"xmlns":2531,"display":2793},[2533,29232,29233,29312],{},[2536,29234,29235,29238,29240],{},[6167,29236,29237],{},"energy error",[2689,29239,573],{},[9491,29241,29242,29286],{},[2536,29243,29244,29246,29248,29250,29252,29254,29256,29258,29260,29262,29264,29270,29272,29274,29276,29278,29280,29282,29284],{},[2689,29245,4425],{"stretchy":2755},[2542,29247,3793],{},[2542,29249,2749],{"mathvariant":2748},[2542,29251,22278],{},[2689,29253,580],{"stretchy":2755},[2542,29255,3138],{},[2689,29257,3649],{"stretchy":2755},[2689,29259,580],{"stretchy":2755},[2542,29261,3138],{},[2689,29263,2691],{},[3168,29265,29266,29268],{},[2542,29267,6182],{},[2693,29269,601],{},[2689,29271,3649],{"stretchy":2755},[2542,29273,22278],{},[2689,29275,580],{"stretchy":2755},[2542,29277,3138],{},[2689,29279,3649],{"stretchy":2755},[2542,29281,2749],{"mathvariant":2748},[2542,29283,3793],{},[2689,29285,2756],{"stretchy":2755},[2536,29287,29288,29290,29292,29294,29296,29298,29300,29306,29308,29310],{},[2689,29289,4425],{"stretchy":2755},[2542,29291,3793],{},[2542,29293,2749],{"mathvariant":2748},[2542,29295,22278],{},[2689,29297,580],{"stretchy":2755},[2542,29299,3138],{},[2539,29301,29302,29304],{},[2689,29303,3649],{"stretchy":2755},[2693,29305,584],{},[2542,29307,2749],{"mathvariant":2748},[2542,29309,3793],{},[2689,29311,2756],{"stretchy":2755},[2549,29313,29314],{"encoding":2551},"\\text{energy error} = \\frac{\\langle\\psi|f(H)(H-E_0)f(H)|\\psi\\rangle}{\\langle\\psi|f(H)^2|\\psi\\rangle}",[507,29316,29318,29339],{"className":29317,"ariaHidden":2557},[2556],[507,29319,29321,29324,29330,29333,29336],{"className":29320},[2561],[507,29322],{"className":29323,"style":28566},[2565],[507,29325,29327],{"className":29326},[2570,7039],[507,29328,29237],{"className":29329},[2570],[507,29331],{"className":29332,"style":2919},[2714],[507,29334,573],{"className":29335},[2923],[507,29337],{"className":29338,"style":2919},[2714],[507,29340,29342,29346],{"className":29341},[2561],[507,29343],{"className":29344,"style":29345},[2565],"height:2.363em;vertical-align:-0.936em;",[507,29347,29349,29352,29559],{"className":29348},[2570],[507,29350],{"className":29351},[2941,9793],[507,29353,29355],{"className":29354},[9491],[507,29356,29358,29550],{"className":29357},[2583,3200],[507,29359,29361,29547],{"className":29360},[2587],[507,29362,29365,29431,29439],{"className":29363,"style":29364},[2591],"height:1.427em;",[507,29366,29367,29370],{"style":11717},[507,29368],{"className":29369,"style":4310},[2599],[507,29371,29373,29376,29379,29382,29385,29388,29391,29422,29425,29428],{"className":29372},[2570],[507,29374,4425],{"className":29375},[2941],[507,29377,3793],{"className":29378,"style":2776},[2570,2611],[507,29380,2749],{"className":29381},[2570],[507,29383,22278],{"className":29384,"style":27338},[2570,2611],[507,29386,580],{"className":29387},[2941],[507,29389,3138],{"className":29390,"style":3153},[2570,2611],[507,29392,29394,29397],{"className":29393},[2780],[507,29395,3649],{"className":29396},[2780],[507,29398,29400],{"className":29399},[2579],[507,29401,29403],{"className":29402},[2583],[507,29404,29406],{"className":29405},[2587],[507,29407,29410],{"className":29408,"style":29409},[2591],"height:0.7401em;",[507,29411,29413,29416],{"style":29412},"top:-2.989em;margin-right:0.05em;",[507,29414],{"className":29415,"style":2600},[2599],[507,29417,29419],{"className":29418},[2604,2605,2606,2607],[507,29420,584],{"className":29421},[2570,2607],[507,29423,2749],{"className":29424},[2570],[507,29426,3793],{"className":29427,"style":2776},[2570,2611],[507,29429,2756],{"className":29430},[2780],[507,29432,29433,29436],{"style":9878},[507,29434],{"className":29435,"style":4310},[2599],[507,29437],{"className":29438,"style":9886},[9885],[507,29440,29441,29444],{"style":9889},[507,29442],{"className":29443,"style":4310},[2599],[507,29445,29447,29450,29453,29456,29459,29462,29465,29468,29471,29474,29477,29480,29483,29523,29526,29529,29532,29535,29538,29541,29544],{"className":29446},[2570],[507,29448,4425],{"className":29449},[2941],[507,29451,3793],{"className":29452,"style":2776},[2570,2611],[507,29454,2749],{"className":29455},[2570],[507,29457,22278],{"className":29458,"style":27338},[2570,2611],[507,29460,580],{"className":29461},[2941],[507,29463,3138],{"className":29464,"style":3153},[2570,2611],[507,29466,3649],{"className":29467},[2780],[507,29469,580],{"className":29470},[2941],[507,29472,3138],{"className":29473,"style":3153},[2570,2611],[507,29475],{"className":29476,"style":2715},[2714],[507,29478,2691],{"className":29479},[2719],[507,29481],{"className":29482,"style":2715},[2714],[507,29484,29486,29489],{"className":29485},[2570],[507,29487,6182],{"className":29488,"style":5892},[2570,2611],[507,29490,29492],{"className":29491},[2579],[507,29493,29495,29515],{"className":29494},[2583,3200],[507,29496,29498,29512],{"className":29497},[2587],[507,29499,29501],{"className":29500,"style":14281},[2591],[507,29502,29503,29506],{"style":6014},[507,29504],{"className":29505,"style":2600},[2599],[507,29507,29509],{"className":29508},[2604,2605,2606,2607],[507,29510,601],{"className":29511},[2570,2607],[507,29513,3225],{"className":29514},[3224],[507,29516,29518],{"className":29517},[2587],[507,29519,29521],{"className":29520,"style":3232},[2591],[507,29522],{},[507,29524,3649],{"className":29525},[2780],[507,29527,22278],{"className":29528,"style":27338},[2570,2611],[507,29530,580],{"className":29531},[2941],[507,29533,3138],{"className":29534,"style":3153},[2570,2611],[507,29536,3649],{"className":29537},[2780],[507,29539,2749],{"className":29540},[2570],[507,29542,3793],{"className":29543,"style":2776},[2570,2611],[507,29545,2756],{"className":29546},[2780],[507,29548,3225],{"className":29549},[3224],[507,29551,29553],{"className":29552},[2587],[507,29554,29557],{"className":29555,"style":29556},[2591],"height:0.936em;",[507,29558],{},[507,29560],{"className":29561},[2780,9793],[507,29563,29565],{"className":29564},[2784],[507,29566,29568,29696],{"className":29567},[2523],[507,29569,29571],{"className":29570},[2527],[2529,29572,29573],{"xmlns":2531,"display":2793},[2533,29574,29575,29693],{},[2536,29576,29577,29579,29691],{},[2689,29578,573],{},[9491,29580,29581,29645],{},[2536,29582,29583,29597,29599,29605,29611,29613,29615,29621,29627,29629,29635,29637,29643],{},[28213,29584,29585,29587,29595],{},[2689,29586,7527],{},[253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\\frac{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2}.",[507,29697,29699,29711],{"className":29698,"ariaHidden":2557},[2556],[507,29700,29702,29705,29708],{"className":29701},[2561],[507,29703],{"className":29704,"style":4728},[2565],[507,29706,573],{"className":29707},[2923],[507,29709],{"className":29710,"style":2919},[2714],[507,29712,29714,29718,30299],{"className":29713},[2561],[507,29715],{"className":29716,"style":29717},[2565],"height:2.8419em;vertical-align:-1.1709em;",[507,29719,29721,29724,30296],{"className":29720},[2570],[507,29722],{"className":29723},[2941,9793],[507,29725,29727],{"className":29726},[9491],[507,29728,29730,30287],{"className":29729},[2583,3200],[507,29731,29733,30284],{"className":29732},[2587],[507,29734,29737,29959,29967],{"className":29735,"style":29736},[2591],"height:1.6709em;",[507,29738,29740,29743],{"style":29739},"top:-2.1288em;",[507,29741],{"className":29742,"style":4310},[2599],[507,29744,29746,29809,29812,29815,29855,29884,29887,29890,29930],{"className":29745},[2570],[507,29747,29749,29752],{"className":29748},[7570],[507,29750,7527],{"className":29751,"style":7576},[7570,7574,7575],[507,29753,29755],{"className":29754},[2579],[507,29756,29758,29801],{"className":29757},[2583,3200],[507,29759,29761,29798],{"className":29760},[2587],[507,29762,29764,29784],{"className":29763,"style":7822},[2591],[507,29765,29766,29769],{"style":7592},[507,29767],{"className":29768,"style":2600},[2599],[507,29770,29772],{"className":29771},[2604,2605,2606,2607],[507,29773,29775,29778,29781],{"className":29774},[2570,2607],[507,29776,3626],{"className":29777,"style":3692},[2570,2611,2607],[507,29779,573],{"className":29780},[2923,2607],[507,29782,601],{"className":29783},[2570,2607],[507,29785,29786,29789],{"style":7845},[507,29787],{"className":29788,"style":2600},[2599],[507,29790,29792],{"className":29791},[2604,2605,2606,2607],[507,29793,29795],{"className":29794},[2570,2607],[507,29796,7681],{"className":29797,"style":3314},[2570,2611,2607],[507,29799,3225],{"className":29800},[3224],[507,29802,29804],{"className":29803},[2587],[507,29805,29807],{"className":29806,"style":7611},[2591],[507,29808],{},[507,29810],{"className":29811,"style":2965},[2714],[507,29813,2749],{"className":29814},[2570],[507,29816,29818,29821],{"className":29817},[2570],[507,29819,28231],{"className":29820,"style":28367},[2570,2611],[507,29822,29824],{"className":29823},[2579],[507,29825,29827,29847],{"className":29826},[2583,3200],[507,29828,29830,29844],{"className":29829},[2587],[507,29831,29833],{"className":29832,"style":11243},[2591],[507,29834,29835,29838],{"style":28382},[507,29836],{"className":29837,"style":2600},[2599],[507,29839,29841],{"className":29840},[2604,2605,2606,2607],[507,29842,3626],{"className":29843,"style":3692},[2570,2611,2607],[507,29845,3225],{"className":29846},[3224],[507,29848,29850],{"className":29849},[2587],[507,29851,29853],{"className":29852,"style":3232},[2591],[507,29854],{},[507,29856,29858,29861],{"className":29857},[2570],[507,29859,2749],{"className":29860},[2570],[507,29862,29864],{"className":29863},[2579],[507,29865,29867],{"className":29866},[2583],[507,29868,29870],{"className":29869},[2587],[507,29871,29873],{"className":29872,"style":29409},[2591],[507,29874,29875,29878],{"style":29412},[507,29876],{"className":29877,"style":2600},[2599],[507,29879,29881],{"className":29880},[2604,2605,2606,2607],[507,29882,584],{"className":29883},[2570,2607],[507,29885,22278],{"className":29886,"style":27338},[2570,2611],[507,29888,580],{"className":29889},[2941],[507,29891,29893,29896],{"className":29892},[2570],[507,29894,6182],{"className":29895,"style":5892},[2570,2611],[507,29897,29899],{"className":29898},[2579],[507,29900,29902,29922],{"className":29901},[2583,3200],[507,29903,29905,29919],{"className":29904},[2587],[507,29906,29908],{"className":29907,"style":11243},[2591],[507,29909,29910,29913],{"style":6014},[507,29911],{"className":29912,"style":2600},[2599],[507,29914,29916],{"className":29915},[2604,2605,2606,2607],[507,29917,3626],{"className":29918,"style":3692},[2570,2611,2607],[507,29920,3225],{"className":29921},[3224],[507,29923,29925],{"className":29924},[2587],[507,29926,29928],{"className":29927,"style":3232},[2591],[507,29929],{},[507,29931,29933,29936],{"className":29932},[2780],[507,29934,3649],{"className":29935},[2780],[507,29937,29939],{"className":29938},[2579],[507,29940,29942],{"className":29941},[2583],[507,29943,29945],{"className":29944},[2587],[507,29946,29948],{"className":29947,"style":29409},[2591],[507,29949,29950,29953],{"style":29412},[507,29951],{"className":29952,"style":2600},[2599],[507,29954,29956],{"className":29955},[2604,2605,2606,2607],[507,29957,584],{"className":29958},[2570,2607],[507,29960,29961,29964],{"style":9878},[507,29962],{"className":29963,"style":4310},[2599],[507,29965],{"className":29966,"style":9886},[9885],[507,29968,29970,29973],{"style":29969},"top:-3.6897em;",[5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turn this into an upper bound that is easier to understand, we first separate the sum in the numerator into terms with 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the first step follows because 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for every 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in the sum, and the second step follows because the sum in the numerator is a subset of the sum in the denominator. For the second term, first we lower bound the denominator by ",[507,33988,33990,34016],{"className":33989},[2523],[507,33991,33993],{"className":33992},[2527],[2529,33994,33995],{"xmlns":2531},[2533,33996,33997,34013],{},[2536,33998,33999,34001,34007],{},[2542,34000,2749],{"mathvariant":2748},[3168,34002,34003,34005],{},[2542,34004,28231],{},[2693,34006,601],{},[2539,34008,34009,34011],{},[2542,34010,2749],{"mathvariant":2748},[2693,34012,584],{},[2549,34014,34015],{"encoding":2551},"|\\gamma_0|^2",[507,34017,34019],{"className":34018,"ariaHidden":2557},[2556],[507,34020,34022,34025,34028,34068],{"className":34021},[2561],[507,34023],{"className":34024,"style":13202},[2565],[507,34026,2749],{"className":34027},[2570],[507,34029,34031,34034],{"className":34030},[2570],[507,34032,28231],{"className":34033,"style":28367},[2570,2611],[507,34035,34037],{"className":34036},[2579],[507,34038,34040,34060],{"className":34039},[2583,3200],[507,34041,34043,34057],{"className":34042},[2587],[507,34044,34046],{"className":34045,"style":14281},[2591],[507,34047,34048,34051],{"style":28382},[507,34049],{"className":34050,"style":2600},[2599],[507,34052,34054],{"className":34053},[2604,2605,2606,2607],[507,34055,601],{"className":34056},[2570,2607],[507,34058,3225],{"className":34059},[3224],[507,34061,34063],{"className":34062},[2587],[507,34064,34066],{"className":34065,"style":3232},[2591],[507,34067],{},[507,34069,34071,34074],{"className":34070},[2570],[507,34072,2749],{"className":34073},[2570],[507,34075,34077],{"className":34076},[2579],[507,34078,34080],{"className":34079},[2583],[507,34081,34083],{"className":34082},[2587],[507,34084,34086],{"className":34085,"style":13224},[2591],[507,34087,34088,34091],{"style":2595},[507,34089],{"className":34090,"style":2600},[2599],[507,34092,34094],{"className":34093},[2604,2605,2606,2607],[507,34095,584],{"className":34096},[2570,2607],", since ",[507,34099,34101,34133],{"className":34100},[2523],[507,34102,34104],{"className":34103},[2527],[2529,34105,34106],{"xmlns":2531},[2533,34107,34108,34130],{},[2536,34109,34110,34112,34114,34120,34126,34128],{},[2542,34111,22278],{},[2689,34113,580],{"stretchy":2755},[3168,34115,34116,34118],{},[2542,34117,6182],{},[2693,34119,601],{},[2539,34121,34122,34124],{},[2689,34123,3649],{"stretchy":2755},[2693,34125,584],{},[2689,34127,573],{},[2693,34129,625],{},[2549,34131,34132],{"encoding":2551},"f(E_0)^2=1",[507,34134,34136,34226],{"className":34135,"ariaHidden":2557},[2556],[507,34137,34139,34142,34145,34148,34188,34217,34220,34223],{"className":34138},[2561],[507,34140],{"className":34141,"style":13202},[2565],[507,34143,22278],{"className":34144,"style":27338},[2570,2611],[507,34146,580],{"className":34147},[2941],[507,34149,34151,34154],{"className":34150},[2570],[507,34152,6182],{"className":34153,"style":5892},[2570,2611],[507,34155,34157],{"className":34156},[2579],[507,34158,34160,34180],{"className":34159},[2583,3200],[507,34161,34163,34177],{"className":34162},[2587],[507,34164,34166],{"className":34165,"style":14281},[2591],[507,34167,34168,34171],{"style":6014},[507,34169],{"className":34170,"style":2600},[2599],[507,34172,34174],{"className":34173},[2604,2605,2606,2607],[507,34175,601],{"className":34176},[2570,2607],[507,34178,3225],{"className":34179},[3224],[507,34181,34183],{"className":34182},[2587],[507,34184,34186],{"className":34185,"style":3232},[2591],[507,34187],{},[507,34189,34191,34194],{"className":34190},[2780],[507,34192,3649],{"className":34193},[2780],[507,34195,34197],{"className":34196},[2579],[507,34198,34200],{"className":34199},[2583],[507,34201,34203],{"className":34202},[2587],[507,34204,34206],{"className":34205,"style":13224},[2591],[507,34207,34208,34211],{"style":2595},[507,34209],{"className":34210,"style":2600},[2599],[507,34212,34214],{"className":34213},[2604,2605,2606,2607],[507,34215,584],{"className":34216},[2570,2607],[507,34218],{"className":34219,"style":2919},[2714],[507,34221,573],{"className":34222},[2923],[507,34224],{"className":34225,"style":2919},[2714],[507,34227,34229,34232],{"className":34228},[2561],[507,34230],{"className":34231,"style":2729},[2565],[507,34233,625],{"className":34234},[2570],": adding everything back together, this gives",[507,34237,34239],{"className":34238},[2784],[507,34240,34242,34356],{"className":34241},[2523],[507,34243,34245],{"className":34244},[2527],[2529,34246,34247],{"xmlns":2531,"display":2793},[2533,34248,34249,34353],{},[2536,34250,34251,34253,34255,34257,34259,34279,34303,34305,34311,34317,34319,34321,34327,34333,34335,34341,34343,34349,34351],{},[6167,34252,29237],{},[2689,34254,27509],{},[2542,34256,27524],{},[2689,34258,2107],{},[9491,34260,34261,34263],{},[2693,34262,625],{},[2536,34264,34265,34267,34273],{},[2542,34266,2749],{"mathvariant":2748},[3168,34268,34269,34271],{},[2542,34270,28231],{},[2693,34272,601],{},[2539,34274,34275,34277],{},[2542,34276,2749],{"mathvariant":2748},[2693,34278,584],{},[30640,34280,34281,34283],{},[2689,34282,7527],{},[2536,34284,34285,34291,34293,34299,34301],{},[3168,34286,34287,34289],{},[2542,34288,6182],{},[2542,34290,3626],{},[2689,34292,1651],{},[3168,34294,34295,34297],{},[2542,34296,6182],{},[2693,34298,601],{},[2689,34300,2107],{},[2542,34302,27524],{},[2542,34304,2749],{"mathvariant":2748},[3168,34306,34307,34309],{},[2542,34308,28231],{},[2542,34310,3626],{},[2539,34312,34313,34315],{},[2542,34314,2749],{"mathvariant":2748},[2693,34316,584],{},[2542,34318,22278],{},[2689,34320,580],{"stretchy":2755},[3168,34322,34323,34325],{},[2542,34324,6182],{},[2542,34326,3626],{},[2539,34328,34329,34331],{},[2689,34330,3649],{"stretchy":2755},[2693,34332,584],{},[2689,34334,580],{"stretchy":2755},[3168,34336,34337,34339],{},[2542,34338,6182],{},[2542,34340,3626],{},[2689,34342,2691],{},[3168,34344,34345,34347],{},[2542,34346,6182],{},[2693,34348,601],{},[2689,34350,3649],{"stretchy":2755},[2542,34352,53],{"mathvariant":2748},[2549,34354,34355],{"encoding":2551},"\\text{energy error} \\le \\delta + \\frac{1}{|\\gamma_0|^2}\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0).",[507,34357,34359,34381,34400,34878],{"className":34358,"ariaHidden":2557},[2556],[507,34360,34362,34366,34372,34375,34378],{"className":34361},[2561],[507,34363],{"className":34364,"style":34365},[2565],"height:0.8304em;vertical-align:-0.1944em;",[507,34367,34369],{"className":34368},[2570,7039],[507,34370,29237],{"className":34371},[2570],[507,34373],{"className":34374,"style":2919},[2714],[507,34376,27509],{"className":34377},[2923],[507,34379],{"className":34380,"style":2919},[2714],[507,34382,34384,34388,34391,34394,34397],{"className":34383},[2561],[507,34385],{"className":34386,"style":34387},[2565],"height:0.7778em;vertical-align:-0.0833em;",[507,34389,27524],{"className":34390,"style":27609},[2570,2611],[507,34392],{"className":34393,"style":2715},[2714],[507,34395,2107],{"className":34396},[2719],[507,34398],{"className":34399,"style":2715},[2714],[507,34401,34403,34407,34538,34541,34676,34679,34682,34722,34751,34754,34757,34797,34826,34829,34869,34872,34875],{"className":34402},[2561],[507,34404],{"className":34405,"style":34406},[2565],"height:2.7294em;vertical-align:-1.408em;",[507,34408,34410,34413,34535],{"className":34409},[2570],[507,34411],{"className":34412},[2941,9793],[507,34414,34416],{"className":34415},[9491],[507,34417,34419,34527],{"className":34418},[2583,3200],[507,34420,34422,34524],{"className":34421},[2587],[507,34423,34425,34505,34513],{"className":34424,"style":9806},[2591],[507,34426,34427,34430],{"style":11717},[507,34428],{"className":34429,"style":4310},[2599],[507,34431,34433,34436,34476],{"className":34432},[2570],[507,34434,2749],{"className":34435},[2570],[507,34437,34439,34442],{"className":34438},[2570],[507,34440,28231],{"className":34441,"style":28367},[2570,2611],[507,34443,34445],{"className":34444},[2579],[507,34446,34448,34468],{"className":34447},[2583,3200],[507,34449,34451,34465],{"className":34450},[2587],[507,34452,34454],{"className":34453,"style":14281},[2591],[507,34455,34456,34459],{"style":28382},[507,34457],{"className":34458,"style":2600},[2599],[507,34460,34462],{"className":34461},[2604,2605,2606,2607],[507,34463,601],{"className":34464},[2570,2607],[507,34466,3225],{"className":34467},[3224],[507,34469,34471],{"className":34470},[2587],[507,34472,34474],{"className":34473,"style":3232},[2591],[507,34475],{},[507,34477,34479,34482],{"className":34478},[2570],[507,34480,2749],{"className":34481},[2570],[507,34483,34485],{"className":34484},[2579],[507,34486,34488],{"className":34487},[2583],[507,34489,34491],{"className":34490},[2587],[507,34492,34494],{"className":34493,"style":29409},[2591],[507,34495,34496,34499],{"style":29412},[507,34497],{"className":34498,"style":2600},[2599],[507,34500,34502],{"className":34501},[2604,2605,2606,2607],[507,34503,584],{"className":34504},[2570,2607],[507,34506,34507,34510],{"style":9878},[507,34508],{"className":34509,"style":4310},[2599],[507,34511],{"className":34512,"style":9886},[9885],[507,34514,34515,34518],{"style":9889},[507,34516],{"className":34517,"style":4310},[2599],[507,34519,34521],{"className":34520},[2570],[507,34522,625],{"className":34523},[2570],[507,34525,3225],{"className":34526},[3224],[507,34528,34530],{"className":34529},[2587],[507,34531,34533],{"className":34532,"style":29556},[2591],[507,34534],{},[507,34536],{"className":34537},[2780,9793],[507,34539],{"className":34540,"style":2965},[2714],[507,34542,34544],{"className":34543},[7570,28286],[507,34545,34547,34667],{"className":34546},[2583,3200],[507,34548,34550,34664],{"className":34549},[2587],[507,34551,34554,34654],{"className":34552,"style":34553},[2591],"height:1.05em;",[507,34555,34556,34559],{"style":28299},[507,34557],{"className":34558,"style":28303},[2599],[507,34560,34562],{"className":34561},[2604,2605,2606,2607],[507,34563,34565,34605,34608,34648,34651],{"className":34564},[2570,2607],[507,34566,34568,34571],{"className":34567},[2570,2607],[507,34569,6182],{"className":34570,"style":5892},[2570,2611,2607],[507,34572,34574],{"className":34573},[2579],[507,34575,34577,34597],{"className":34576},[2583,3200],[507,34578,34580,34594],{"className":34579},[2587],[507,34581,34583],{"className":34582,"style":31223},[2591],[507,34584,34585,34588],{"style":31226},[507,34586],{"className":34587,"style":4812},[2599],[507,34589,34591],{"className":34590},[2604,4816,2948,2607],[507,34592,3626],{"className":34593,"style":3692},[2570,2611,2607],[507,34595,3225],{"className":34596},[3224],[507,34598,34600],{"className":34599},[2587],[507,34601,34603],{"className":34602,"style":31245},[2591],[507,34604],{},[507,34606,1651],{"className":34607},[2923,2607],[507,34609,34611,34614],{"className":34610},[2570,2607],[507,34612,6182],{"className":34613,"style":5892},[2570,2611,2607],[507,34615,34617],{"className":34616},[2579],[507,34618,34620,34640],{"className":34619},[2583,3200],[507,34621,34623,34637],{"className":34622},[2587],[507,34624,34626],{"className":34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sName":34915},[3224],[507,34917,34919],{"className":34918},[2587],[507,34920,34922],{"className":34921,"style":3232},[2591],[507,34923],{},[507,34925,3649],{"className":34926},[2780],[507,34928,53],{"className":34929},[2570],[18,34931,34932,34933,35002,35003,35031,35032,35253,35254,35422,35423,35712],{},"To simplify what is left, notice that for all these 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by the definition of ",[507,35004,35006,35019],{"className":35005},[2523],[507,35007,35009],{"className":35008},[2527],[2529,35010,35011],{"xmlns":2531},[2533,35012,35013,35017],{},[2536,35014,35015],{},[2542,35016,22278],{},[2549,35018,22278],{"encoding":2551},[507,35020,35022],{"className":35021,"ariaHidden":2557},[2556],[507,35023,35025,35028],{"className":35024},[2561],[507,35026],{"className":35027,"style":7035},[2565],[507,35029,22278],{"className":35030,"style":27338},[2570,2611]," we know that ",[507,35033,35035,35089],{"className":35034},[2523],[507,35036,35038],{"className":35037},[2527],[2529,35039,35040],{"xmlns":2531},[2533,35041,35042,35086],{},[2536,35043,35044,35046,35048,35054,35060,35062,35064],{},[2542,35045,22278],{},[2689,35047,580],{"stretchy":2755},[3168,35049,35050,35052],{},[2542,35051,6182],{},[2542,35053,3626],{},[2539,35055,35056,35058],{},[2689,35057,3649],{"stretchy":2755},[2693,35059,584],{},[2689,35061,27509],{},[2693,35063,12152],{},[2539,35065,35066,35078],{},[2536,35067,35068,35070,35072,35074,35076],{},[2689,35069,580],{"fence":2557},[2693,35071,625],{},[2689,35073,2107],{},[2542,35075,27524],{},[2689,35077,3649],{"fence":2557},[2536,35079,35080,35082,35084],{},[2689,35081,2691],{},[2693,35083,584],{},[2542,35085,4959],{},[2549,35087,35088],{"encoding":2551},"f(E_k)^2 \\le 4\\left(1 + \\delta\\right)^{-2d}",[507,35090,35092,35182],{"className":35091,"ariaHidden":2557},[2556],[507,35093,35095,35098,35101,35104,35144,35173,35176,35179],{"className":35094},[2561],[507,35096],{"className":35097,"style":13202},[2565],[507,35099,22278],{"className":35100,"style":27338},[2570,2611],[507,35102,580],{"className":35103},[2941],[507,35105,35107,35110],{"className":35106},[2570],[507,35108,6182],{"className":35109,"style":5892},[2570,2611],[507,35111,35113],{"className":35112},[2579],[507,35114,35116,35136],{"className":35115},[2583,3200],[507,35117,35119,35133],{"className":35118},[2587],[507,35120,35122],{"className":35121,"style":11243},[2591],[507,35123,35124,35127],{"style":6014},[507,35125],{"className":35126,"style":2600},[2599],[507,35128,35130],{"className":35129},[2604,2605,2606,2607],[507,35131,3626],{"className":35132,"style":3692},[2570,2611,2607],[507,35134,3225],{"className":35135},[3224],[507,35137,35139],{"className":35138},[2587],[507,35140,35142],{"className":35141,"style":3232},[2591],[507,35143],{},[507,35145,35147,35150],{"className":35146},[2780],[507,35148,3649],{"className":35149},[2780],[507,35151,35153],{"className":35152},[2579],[507,35154,35156],{"className":35155},[2583],[507,35157,35159],{"className":35158},[2587],[507,35160,35162],{"className":35161,"style":13224},[2591],[507,35163,35164,35167],{"style":2595},[507,35165],{"className":35166,"style":2600},[2599],[507,35168,35170],{"className":35169},[2604,2605,2606,2607],[507,35171,584],{"className":35172},[2570,2607],[507,35174],{"className":35175,"style":2919},[2714],[507,35177,27509],{"className":35178},[2923],[507,35180],{"className":35181,"style":2919},[2714],[507,35183,35185,35188,35191,35194],{"className":35184},[2561],[507,35186],{"className":35187,"style":27578},[2565],[507,35189,12152],{"className":35190},[2570],[507,35192],{"className":35193,"style":2965},[2714],[507,35195,35197,35221],{"className":35196},[2937],[507,35198,35200,35203,35206,35209,35212,35215,35218],{"className":35199},[2937],[507,35201,580],{"className":35202,"style":2943},[2941,2942],[507,35204,625],{"className":35205},[2570],[507,35207],{"className":35208,"style":2715},[2714],[507,35210,2107],{"className":35211},[2719],[507,35213],{"className":35214,"style":2715},[2714],[507,35216,27524],{"className":35217,"style":27609},[2570,2611],[507,35219,3649],{"className":35220,"style":2943},[2780,2942],[507,35222,35224],{"className":35223},[2579],[507,35225,35227],{"className":35226},[2583],[507,35228,35230],{"className":35229},[2587],[507,35231,35233],{"className":35232,"style":27625},[2591],[507,35234,35235,35238],{"style":7845},[507,35236],{"className":35237,"style":2600},[2599],[507,35239,35241],{"className":35240},[2604,2605,2606,2607],[507,35242,35244,35247,35250],{"className":35243},[2570,2607],[507,35245,2691],{"className":35246},[2570,2607],[507,35248,584],{"className":35249},[2570,2607],[507,35251,4959],{"className":35252},[2570,2611,2607],". Additionally upper bounding ",[507,35255,35257,35293],{"className":35256},[2523],[507,35258,35260],{"className":35259},[2527],[2529,35261,35262],{"xmlns":2531},[2533,35263,35264,35290],{},[2536,35265,35266,35272,35274,35280,35282,35284,35286,35288],{},[3168,35267,35268,35270],{},[2542,35269,6182],{},[2542,35271,3626],{},[2689,35273,2691],{},[3168,35275,35276,35278],{},[2542,35277,6182],{},[2693,35279,601],{},[2689,35281,5677],{},[2693,35283,584],{},[2542,35285,12158],{"mathvariant":2748},[2542,35287,3138],{},[2542,35289,12158],{"mathvariant":2748},[2549,35291,35292],{"encoding":2551},"E_k-E_0\u003C2\\|H\\|",[507,35294,35296,35351,35406],{"className":35295,"ariaHidden":2557},[2556],[507,35297,35299,35302,35342,35345,35348],{"className":35298},[2561],[507,35300],{"className":35301,"style":3187},[2565],[507,35303,35305,35308],{"className":35304},[2570],[507,35306,6182],{"className":35307,"style":5892},[2570,2611],[507,35309,35311],{"className":35310},[2579],[507,35312,35314,35334],{"className":35313},[2583,3200],[507,35315,35317,35331],{"className":35316},[2587],[507,35318,35320],{"className":35319,"style":11243},[2591],[507,35321,35322,35325],{"style":6014},[507,35323],{"className":35324,"style":2600},[2599],[507,35326,35328],{"className":35327},[2604,2605,2606,2607],[507,35329,3626],{"className":35330,"style":3692},[2570,2611,2607],[507,35332,3225],{"className":35333},[3224],[507,35335,35337],{"className":35336},[2587],[507,35338,35340],{"className":35339,"style":3232},[2591],[507,35341],{},[507,35343],{"className":35344,"style":2715},[2714],[507,35346,2691],{"className":35347},[2719],[507,35349],{"className":35350,"style":2715},[2714],[507,35352,35354,35357,35397,35400,35403],{"className":35353},[2561],[507,35355],{"className":35356,"style":3187},[2565],[507,35358,35360,35363],{"className":35359},[2570],[507,35361,6182],{"className":35362,"style":5892},[2570,2611],[507,35364,35366],{"className":35365},[2579],[507,35367,35369,35389],{"className":35368},[2583,3200],[507,35370,35372,35386],{"className":35371},[2587],[507,35373,35375],{"className":35374,"style":14281},[2591],[507,35376,35377,35380],{"style":6014},[507,35378],{"className":35379,"style":2600},[2599],[507,35381,35383],{"className":35382},[2604,2605,2606,2607],[507,35384,601],{"className":35385},[2570,2607],[507,35387,3225],{"className":35388},[3224],[507,35390,35392],{"className":35391},[2587],[507,35393,35395],{"className":35394,"style":3232},[2591],[507,35396],{},[507,35398],{"className":35399,"style":2919},[2714],[507,35401,5677],{"className":35402},[2923],[507,35404],{"className":35405,"style":2919},[2714],[507,35407,35409,35412,35416,35419],{"className":35408},[2561],[507,35410],{"className":35411,"style":2769},[2565],[507,35413,35415],{"className":35414},[2570],"2∥",[507,35417,3138],{"className":35418,"style":3153},[2570,2611],[507,35420,12158],{"className":35421},[2570]," and upper bounding ",[507,35424,35426,35480],{"className":35425},[2523],[507,35427,35429],{"className":35428},[2527],[2529,35430,35431],{"xmlns":2531},[2533,35432,35433,35477],{},[2536,35434,35435,35459,35461,35467,35473,35475],{},[3168,35436,35437,35439],{},[2689,35438,7527],{},[2536,35440,35441,35447,35449,35455,35457],{},[3168,35442,35443,35445],{},[2542,35444,6182],{},[2542,35446,3626],{},[2689,35448,1651],{},[3168,35450,35451,35453],{},[2542,35452,6182],{},[2693,35454,601],{},[2689,35456,2107],{},[2542,35458,27524],{},[2542,35460,2749],{"mathvariant":2748},[3168,35462,35463,35465],{},[2542,35464,28231],{},[2542,35466,3626],{},[2539,35468,35469,35471],{},[2542,35470,2749],{"mathvariant":2748},[2693,35472,584],{},[2689,35474,5677],{},[2693,35476,625],{},[2549,35478,35479],{"encoding":2551},"\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2\u003C1",[507,35481,35483,35703],{"className":35482,"ariaHidden":2557},[2556],[507,35484,35486,35490,35619,35622,35625,35665,35694,35697,35700],{"className":35485},[2561],[507,35487],{"className":35488,"style":35489},[2565],"height:1.2197em;vertical-align:-0.4056em;",[507,35491,35493,35496],{"className":35492},[7570],[507,35494,7527],{"className":35495,"style":7576},[7570,7574,7575],[507,35497,35499],{"className":35498},[2579],[507,35500,35502,35611],{"className":35501},[2583,3200],[507,35503,35505,35608],{"className":35504},[2587],[507,35506,35508],{"className":35507,"style":31193},[2591],[507,35509,35510,35513],{"style":7592},[507,35511],{"className":35512,"style":2600},[2599],[507,35514,35516],{"className":35515},[2604,2605,2606,2607],[507,35517,35519,35559,35562,35602,35605],{"className":35518},[2570,2607],[507,35520,35522,35525],{"className":35521},[2570,2607],[507,35523,6182],{"className":35524,"style":5892},[2570,2611,2607],[507,35526,35528],{"className":35527},[2579],[507,35529,35531,35551],{"className":35530},[2583,3200],[507,35532,35534,35548],{"className":35533},[2587],[507,35535,35537],{"className":35536,"style":31223},[2591],[507,35538,35539,35542],{"style":31226},[507,35540],{"className":35541,"style":4812},[2599],[507,35543,35545],{"className":35544},[2604,4816,2948,2607],[507,35546,3626],{"className":35547,"style":3692},[2570,2611,2607],[507,35549,3225],{"className":35550},[3224],[507,35552,35554],{"className":35553},[2587],[507,35555,35557],{"className":35556,"style":31245},[2591],[507,35558],{},[507,35560,1651],{"className":35561},[2923,2607],[507,35563,35565,35568],{"className":35564},[2570,2607],[507,35566,6182],{"className":35567,"style":5892},[2570,2611,2607],[507,35569,35571],{"className":35570},[2579],[507,35572,35574,35594],{"className":35573},[2583,3200],[507,35575,35577,35591],{"className":35576},[2587],[507,35578,35580],{"className":35579,"style":31269},[2591],[507,35581,35582,35585],{"style":31272},[507,35583],{"className":35584,"style":4812},[2599],[507,35586,35588],{"className":35587},[2604,4816,2948,2607],[507,35589,601],{"className":35590},[2570,2607],[507,35592,3225],{"className":35593},[3224],[507,35595,35597],{"className":35596},[2587],[507,35598,35600],{"className":35599,"style":4829},[2591],[507,35601],{},[507,35603,2107],{"className":35604},[2719,2607],[507,35606,27524],{"className":35607,"style":27609},[2570,2611,2607],[507,35609,3225],{"className":35610},[3224],[507,35612,35614],{"className":35613},[2587],[507,35615,35617],{"className":35616,"style":31308},[2591],[507,35618],{},[507,35620],{"className":35621,"style":2965},[2714],[507,35623,2749],{"className":35624},[2570],[507,35626,35628,35631],{"className":35627},[2570],[507,35629,28231],{"className":35630,"style":28367},[2570,2611],[507,35632,35634],{"className":35633},[2579],[507,35635,35637,35657],{"className":35636},[2583,3200],[507,35638,35640,35654],{"className":35639},[2587],[507,35641,35643],{"className":35642,"style":11243},[2591],[507,35644,35645,35648],{"style":28382},[507,35646],{"className":35647,"style":2600},[2599],[507,35649,35651],{"className":35650},[2604,2605,2606,2607],[507,35652,3626],{"className":35653,"style":3692},[2570,2611,2607],[507,35655,3225],{"className":35656},[3224],[507,35658,35660],{"className":35659},[2587],[507,35661,35663],{"className":35662,"style":3232},[2591],[507,35664],{},[507,35666,35668,35671],{"className":35667},[2570],[507,35669,2749],{"className":35670},[2570],[507,35672,35674],{"className":35673},[2579],[507,35675,35677],{"className":35676},[2583],[507,35678,35680],{"className":35679},[2587],[507,35681,35683],{"className":35682,"style":13224},[2591],[507,35684,35685,35688],{"style":2595},[507,35686],{"className":35687,"style":2600},[2599],[507,35689,35691],{"className":35690},[2604,2605,2606,2607],[507,35692,584],{"className":35693},[2570,2607],[507,35695],{"className":35696,"style":2919},[2714],[507,35698,5677],{"className":35699},[2923],[507,35701],{"className":35702,"style":2919},[2714],[507,35704,35706,35709],{"className":35705},[2561],[507,35707],{"className":35708,"style":2729},[2565],[507,35710,625],{"className":35711},[2570]," gives",[507,35714,35716],{"className":35715},[2784],[507,35717,35719,35790],{"className":35718},[2523],[507,35720,35722],{"className":35721},[2527],[2529,35723,35724],{"xmlns":2531,"display":2793},[2533,35725,35726,35787],{},[2536,35727,35728,35730,35732,35734,35736,35757,35759,35761,35763,35785],{},[6167,35729,29237],{},[2689,35731,27509],{},[2542,35733,27524],{},[2689,35735,2107],{},[9491,35737,35738,35741],{},[2693,35739,35740],{},"8",[2536,35742,35743,35745,35751],{},[2542,35744,2749],{"mathvariant":2748},[3168,35746,35747,35749],{},[2542,35748,28231],{},[2693,35750,601],{},[2539,35752,35753,35755],{},[2542,35754,2749],{"mathvariant":2748},[2693,35756,584],{},[2542,35758,12158],{"mathvariant":2748},[2542,35760,3138],{},[2542,35762,12158],{"mathvariant":2748},[2539,35764,35765,35777],{},[2536,35766,35767,35769,35771,35773,35775],{},[2689,35768,580],{"fence":2557},[2693,35770,625],{},[2689,35772,2107],{},[2542,35774,27524],{},[2689,35776,3649],{"fence":2557},[2536,35778,35779,35781,35783],{},[2689,35780,2691],{},[2693,35782,584],{},[2542,35784,4959],{},[2542,35786,53],{"mathvariant":2748},[2549,35788,35789],{"encoding":2551},"\\text{energy error} \\le \\delta + \\frac{8}{|\\gamma_0|^2}\\|H\\|\\left(1 + \\delta\\right)^{-2d}.",[507,35791,35793,35814,35832],{"className":35792,"ariaHidden":2557},[2556],[507,35794,35796,35799,35805,35808,35811],{"className":35795},[2561],[507,35797],{"className":35798,"style":34365},[2565],[507,35800,35802],{"className":35801},[2570,7039],[507,35803,29237],{"className":35804},[2570],[507,35806],{"className":35807,"style":2919},[2714],[507,35809,27509],{"className":35810},[2923],[507,35812],{"className":35813,"style":2919},[2714],[507,35815,35817,35820,35823,35826,35829],{"className":35816},[2561],[507,35818],{"className":35819,"style":34387},[2565],[507,35821,27524],{"className":35822,"style":27609},[2570,2611],[507,35824],{"className":35825,"style":2715},[2714],[507,35827,2107],{"className":35828},[2719],[507,35830],{"className":35831,"style":2715},[2714],[507,35833,35835,35839,35970,35973,35976,35979,35982,36041,36044],{"className":35834},[2561],[507,35836],{"className":35837,"style":35838},[2565],"height:2.2574em;vertical-align:-0.936em;",[507,35840,35842,35845,35967],{"className":35841},[2570],[507,35843],{"className":35844},[2941,9793],[507,35846,35848],{"className":35847},[9491],[507,35849,35851,35959],{"className":35850},[2583,3200],[507,35852,35854,35956],{"className":35853},[2587],[507,35855,35857,35937,35945],{"className":35856,"style":9806},[2591],[507,35858,35859,35862],{"style":11717},[507,35860],{"className":35861,"style":4310},[2599],[507,35863,35865,35868,35908],{"className":35864},[2570],[507,35866,2749],{"className":35867},[2570],[507,35869,35871,35874],{"className":35870},[2570],[507,35872,28231],{"className":35873,"style":28367},[2570,2611],[507,35875,35877],{"className":35876},[2579],[507,35878,35880,35900],{"className":35879},[2583,3200],[507,35881,35883,35897],{"className":35882},[2587],[507,35884,35886],{"className":35885,"style":14281},[2591],[507,35887,35888,35891],{"style":28382},[507,35889],{"className":35890,"style":2600},[2599],[507,35892,35894],{"className":35893},[2604,2605,2606,2607],[507,35895,601],{"className":35896},[2570,2607],[507,35898,3225],{"className":35899},[3224],[507,35901,35903],{"className":35902},[2587],[507,35904,35906],{"className":35905,"style":3232},[2591],[507,35907],{},[507,35909,35911,35914],{"className":35910},[2570],[507,35912,2749],{"className":35913},[2570],[507,35915,35917],{"className":35916},[2579],[507,35918,35920],{"className":35919},[2583],[507,35921,35923],{"className":35922},[2587],[507,35924,35926],{"className":35925,"style":29409},[2591],[507,35927,35928,35931],{"style":29412},[507,35929],{"className":35930,"style":2600},[2599],[507,35932,35934],{"className":35933},[2604,2605,2606,2607],[507,35935,584],{"className":35936},[2570,2607],[507,35938,35939,35942],{"style":9878},[507,35940],{"className":35941,"style":4310},[2599],[507,35943],{"className":35944,"style":9886},[9885],[507,35946,35947,35950],{"style":9889},[507,35948],{"className":35949,"style":4310},[2599],[507,35951,35953],{"className":35952},[2570],[507,35954,35740],{"className":35955},[2570],[507,35957,3225],{"className":35958},[3224],[507,35960,35962],{"className":35961},[2587],[507,35963,35965],{"className":35964,"style":29556},[2591],[507,35966],{},[507,35968],{"className":35969},[2780,9793],[507,35971,12158],{"className":35972},[2570],[507,35974,3138],{"className":35975,"style":3153},[2570,2611],[507,35977,12158],{"className":35978},[2570],[507,35980],{"className":35981,"style":2965},[2714],[507,35983,35985,36009],{"className":35984},[2937],[507,35986,35988,35991,35994,35997,36000,36003,36006],{"className":35987},[2937],[507,35989,580],{"className":35990,"style":2943},[2941,2942],[507,35992,625],{"className":35993},[2570],[507,35995],{"className":35996,"style":2715},[2714],[507,35998,2107],{"className":35999},[2719],[507,36001],{"className":36002,"style":2715},[2714],[507,36004,27524],{"className":36005,"style":27609},[2570,2611],[507,36007,3649],{"className":36008,"style":2943},[2780,2942],[507,36010,36012],{"className":36011},[2579],[507,36013,36015],{"className":36014},[2583],[507,36016,36018],{"className":36017},[2587],[507,36019,36021],{"className":36020,"style":27625},[2591],[507,36022,36023,36026],{"style":7845},[507,36024],{"className":36025,"style":2600},[2599],[507,36027,36029],{"className":36028},[2604,2605,2606,2607],[507,36030,36032,36035,36038],{"className":36031},[2570,2607],[507,36033,2691],{"className":36034},[2570,2607],[507,36036,584],{"className":36037},[2570,2607],[507,36039,4959],{"className":36040},[2570,2611,2607],[507,36042],{"className":36043,"style":2965},[2714],[507,36045,53],{"className":36046},[2570],[18,36048,36049,36050,36102,36103,36131,36132,36188,36189,36344,36345,36373],{},"This holds for any ",[507,36051,36053,36071],{"className":36052},[2523],[507,36054,36056],{"className":36055},[2527],[2529,36057,36058],{"xmlns":2531},[2533,36059,36060,36068],{},[2536,36061,36062,36064,36066],{},[2542,36063,27524],{},[2689,36065,1651],{},[2693,36067,601],{},[2549,36069,36070],{"encoding":2551},"\\delta>0",[507,36072,36074,36093],{"className":36073,"ariaHidden":2557},[2556],[507,36075,36077,36081,36084,36087,36090],{"className":36076},[2561],[507,36078],{"className":36079,"style":36080},[2565],"height:0.7335em;vertical-align:-0.0391em;",[507,36082,27524],{"className":36083,"style":27609},[2570,2611],[507,36085],{"className":36086,"style":2919},[2714],[507,36088,1651],{"className":36089},[2923],[507,36091],{"className":36092,"style":2919},[2714],[507,36094,36096,36099],{"className":36095},[2561],[507,36097],{"className":36098,"style":2729},[2565],[507,36100,601],{"className":36101},[2570],", so if we set ",[507,36104,36106,36119],{"className":36105},[2523],[507,36107,36109],{"className":36108},[2527],[2529,36110,36111],{"xmlns":2531},[2533,36112,36113,36117],{},[2536,36114,36115],{},[2542,36116,27524],{},[2549,36118,32250],{"encoding":2551},[507,36120,36122],{"className":36121,"ariaHidden":2557},[2556],[507,36123,36125,36128],{"className":36124},[2561],[507,36126],{"className":36127,"style":5434},[2565],[507,36129,27524],{"className":36130,"style":27609},[2570,2611]," equal to our goal error, then the error bound above converges towards that exponentially with the Krylov dimension ",[507,36133,36135,36155],{"className":36134},[2523],[507,36136,36138],{"className":36137},[2527],[2529,36139,36140],{"xmlns":2531},[2533,36141,36142,36152],{},[2536,36143,36144,36146,36148,36150],{},[2693,36145,584],{},[2542,36147,4959],{},[2689,36149,573],{},[2542,36151,2216],{},[2549,36153,36154],{"encoding":2551},"2d=r",[507,36156,36158,36179],{"className":36157,"ariaHidden":2557},[2556],[507,36159,36161,36164,36167,36170,36173,36176],{"className":36160},[2561],[507,36162],{"className":36163,"style":5434},[2565],[507,36165,584],{"className":36166},[2570],[507,36168,4959],{"className":36169},[2570,2611],[507,36171],{"className":36172,"style":2919},[2714],[507,36174,573],{"className":36175},[2923],[507,36177],{"className":36178,"style":2919},[2714],[507,36180,36182,36185],{"className":36181},[2561],[507,36183],{"className":36184,"style":2639},[2565],[507,36186,2216],{"className":36187,"style":2612},[2570,2611],". Also note that if ",[507,36190,36192,36222],{"className":36191},[2523],[507,36193,36195],{"className":36194},[2527],[2529,36196,36197],{"xmlns":2531},[2533,36198,36199,36219],{},[2536,36200,36201,36203,36205,36211,36213],{},[2542,36202,27524],{},[2689,36204,5677],{},[3168,36206,36207,36209],{},[2542,36208,6182],{},[2693,36210,625],{},[2689,36212,2691],{},[3168,36214,36215,36217],{},[2542,36216,6182],{},[2693,36218,601],{},[2549,36220,36221],{"encoding":2551},"\\delta\u003CE_1-E_0",[507,36223,36225,36243,36298],{"className":36224,"ariaHidden":2557},[2556],[507,36226,36228,36231,36234,36237,36240],{"className":36227},[2561],[507,36229],{"className":36230,"style":36080},[2565],[507,36232,27524],{"className":36233,"style":27609},[2570,2611],[507,36235],{"className":36236,"style":2919},[2714],[507,36238,5677],{"className":36239},[2923],[507,36241],{"className":36242,"style":2919},[2714],[507,36244,36246,36249,36289,36292,36295],{"className":36245},[2561],[507,36247],{"className":36248,"style":3187},[2565],[507,36250,36252,36255],{"className":36251},[2570],[507,36253,6182],{"className":36254,"style":5892},[2570,2611],[507,36256,36258],{"className":36257},[2579],[507,36259,36261,36281],{"className":36260},[2583,3200],[507,36262,36264,36278],{"className":36263},[2587],[507,36265,36267],{"className":36266,"style":14281},[2591],[507,36268,36269,36272],{"style":6014},[507,36270],{"className":36271,"style":2600},[2599],[507,36273,36275],{"className":36274},[2604,2605,2606,2607],[507,36276,625],{"className":36277},[2570,2607],[507,36279,3225],{"className":36280},[3224],[507,36282,36284],{"className":36283},[2587],[507,36285,36287],{"className":36286,"style":3232},[2591],[507,36288],{},[507,36290],{"className":36291,"style":2715},[2714],[507,36293,2691],{"className":36294},[2719],[507,36296],{"className":36297,"style":2715},[2714],[507,36299,36301,36304],{"className":36300},[2561],[507,36302],{"className":36303,"style":3187},[2565],[507,36305,36307,36310],{"className":36306},[2570],[507,36308,6182],{"className":36309,"style":5892},[2570,2611],[507,36311,36313],{"className":36312},[2579],[507,36314,36316,36336],{"className":36315},[2583,3200],[507,36317,36319,36333],{"className":36318},[2587],[507,36320,36322],{"className":36321,"style":14281},[2591],[507,36323,36324,36327],{"style":6014},[507,36325],{"className":36326,"style":2600},[2599],[507,36328,36330],{"className":36329},[2604,2605,2606,2607],[507,36331,601],{"className":36332},[2570,2607],[507,36334,3225],{"className":36335},[3224],[507,36337,36339],{"className":36338},[2587],[507,36340,36342],{"className":36341,"style":3232},[2591],[507,36343],{}," then the ",[507,36346,36348,36361],{"className":36347},[2523],[507,36349,36351],{"className":36350},[2527],[2529,36352,36353],{"xmlns":2531},[2533,36354,36355,36359],{},[2536,36356,36357],{},[2542,36358,27524],{},[2549,36360,32250],{"encoding":2551},[507,36362,36364],{"className":36363,"ariaHidden":2557},[2556],[507,36365,36367,36370],{"className":36366},[2561],[507,36368],{"className":36369,"style":5434},[2565],[507,36371,27524],{"className":36372,"style":27609},[2570,2611]," term actually goes away entirely in the above bound.",[18,36375,36376,36377,36435],{},"To complete the argument, we first note that the above is just the energy error of the particular state ",[507,36378,36380,36405],{"className":36379},[2523],[507,36381,36383],{"className":36382},[2527],[2529,36384,36385],{"xmlns":2531},[2533,36386,36387,36403],{},[2536,36388,36389,36391,36393,36395,36397,36399,36401],{},[2542,36390,22278],{},[2689,36392,580],{"stretchy":2755},[2542,36394,3138],{},[2689,36396,3649],{"stretchy":2755},[2542,36398,2749],{"mathvariant":2748},[2542,36400,3793],{},[2689,36402,2756],{"stretchy":2755},[2549,36404,28117],{"encoding":2551},[507,36406,36408],{"className":36407,"ariaHidden":2557},[2556],[507,36409,36411,36414,36417,36420,36423,36426,36429,36432],{"className":36410},[2561],[507,36412],{"className":36413,"style":2769},[2565],[507,36415,22278],{"className":36416,"style":27338},[2570,2611],[507,36418,580],{"className":36419},[2941],[507,36421,3138],{"className":36422,"style":3153},[2570,2611],[507,36424,3649],{"className":36425},[2780],[507,36427,2749],{"className":36428},[2570],[507,36430,3793],{"className":36431,"style":2776},[2570,2611],[507,36433,2756],{"className":36434},[2780],", rather than the energy error of the lowest energy state in the Krylov space. However, by the (Rayleigh-Ritz) variational principle, the energy error of the lowest energy state in the Krylov space is upper bounded by the energy error of any state in the Krylov space, so the above is also an upper bound on the energy error of the lowest energy state, that is, the output of the Krylov quantum diagonalization algorithm.",[18,36437,36438,36439,10799,36441,36444],{},"A similar analysis as the above can be carried out that additionally accounts for noise and the thresholding procedure discussed in the notebook. See ",[49,36440,8547],{"href":1086},[49,36442,36443],{"href":1086},"[4]"," for this analysis.",[13,36446,36448],{"id":36447},"appendix-proof-of-claim-1","Appendix: proof of Claim 1",[18,36450,36451,36452,36454,36455,36529,36530,36618,36619,36647],{},"The following is mostly derived from ",[49,36453,27303],{"href":1086},", Theorem 3.1: let ",[507,36456,36458,36480],{"className":36457},[2523],[507,36459,36461],{"className":36460},[2527],[2529,36462,36463],{"xmlns":2531},[2533,36464,36465,36477],{},[2536,36466,36467,36469,36471,36473,36475],{},[2693,36468,601],{},[2689,36470,5677],{},[2542,36472,49],{},[2689,36474,5677],{},[2542,36476,13102],{},[2549,36478,36479],{"encoding":2551},"0 \u003C a \u003C b",[507,36481,36483,36502,36520],{"className":36482,"ariaHidden":2557},[2556],[507,36484,36486,36490,36493,36496,36499],{"className":36485},[2561],[507,36487],{"className":36488,"style":36489},[2565],"height:0.6835em;vertical-align:-0.0391em;",[507,36491,601],{"className":36492},[2570],[507,36494],{"className":36495,"style":2919},[2714],[507,36497,5677],{"className":36498},[2923],[507,36500],{"className":36501,"style":2919},[2714],[507,36503,36505,36508,36511,36514,36517],{"className":36504},[2561],[507,36506],{"className":36507,"style":5695},[2565],[507,36509,49],{"className":36510},[2570,2611],[507,36512],{"className":36513,"style":2919},[2714],[507,36515,5677],{"className":36516},[2923],[507,36518],{"className":36519,"style":2919},[2714],[507,36521,36523,36526],{"className":36522},[2561],[507,36524],{"className":36525,"style":5434},[2565],[507,36527,13102],{"className":36528},[2570,2611]," and let ",[507,36531,36533,36554],{"className":36532},[2523],[507,36534,36536],{"className":36535},[2527],[2529,36537,36538],{"xmlns":2531},[2533,36539,36540,36551],{},[2536,36541,36542],{},[3775,36543,36544,36547,36549],{},[2542,36545,36546],{"mathvariant":2748},"Π",[2542,36548,4959],{},[2689,36550,20769],{},[2549,36552,36553],{"encoding":2551},"\\Pi^*_d",[507,36555,36557],{"className":36556,"ariaHidden":2557},[2556],[507,36558,36560,36564],{"className":36559},[2561],[507,36561],{"className":36562,"style":36563},[2565],"height:0.9718em;vertical-align:-0.2831em;",[507,36565,36567,36570],{"className":36566},[2570],[507,36568,36546],{"className":36569},[2570],[507,36571,36573],{"className":36572},[2579],[507,36574,36576,36609],{"className":36575},[2583,3200],[507,36577,36579,36606],{"className":36578},[2587],[507,36580,36583,36595],{"className":36581,"style":36582},[2591],"height:0.6887em;",[507,36584,36586,36589],{"style":36585},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[507,36587],{"className":36588,"style":2600},[2599],[507,36590,36592],{"className":36591},[2604,2605,2606,2607],[507,36593,4959],{"className":36594},[2570,2611,2607],[507,36596,36597,36600],{"style":2595},[507,36598],{"className":36599,"style":2600},[2599],[507,36601,36603],{"className":36602},[2604,2605,2606,2607],[507,36604,20769],{"className":36605},[2719,2607],[507,36607,3225],{"className":36608},[3224],[507,36610,36612],{"className":36611},[2587],[507,36613,36616],{"className":36614,"style":36615},[2591],"height:0.2831em;",[507,36617],{}," be the space of residual polynomials (polynomials whose value at 0 is 1) of degree at most ",[507,36620,36622,36635],{"className":36621},[2523],[507,36623,36625],{"className":36624},[2527],[2529,36626,36627],{"xmlns":2531},[2533,36628,36629,36633],{},[2536,36630,36631],{},[2542,36632,4959],{},[2549,36634,4959],{"encoding":2551},[507,36636,36638],{"className":36637,"ariaHidden":2557},[2556],[507,36639,36641,36644],{"className":36640},[2561],[507,36642],{"className":36643,"style":5434},[2565],[507,36645,4959],{"className":36646},[2570,2611],". The solution to",[507,36649,36651],{"className":36650},[2784],[507,36652,36654,36748],{"className":36653},[2523],[507,36655,36657],{"className":36656},[2527],[2529,36658,36659],{"xmlns":2531,"display":2793},[2533,36660,36661,36745],{},[2536,36662,36663,36666,36668,36670,36672,36674,36676,36678,36680,36682,36706,36731,36733,36735,36737,36739,36741,36743],{},[2542,36664,36665],{},"β",[2689,36667,580],{"stretchy":2755},[2542,36669,49],{},[2689,36671,2819],{"separator":2557},[2542,36673,13102],{},[2689,36675,2819],{"separator":2557},[2542,36677,4959],{},[2689,36679,3649],{"stretchy":2755},[2689,36681,573],{},[30640,36683,36684,36691],{},[2536,36685,36686,36688],{},[2542,36687,25230],{},[2689,36689,36690],{},"⁡",[2536,36692,36693,36695,36698],{},[2542,36694,18],{},[2689,36696,36697],{},"∈",[3775,36699,36700,36702,36704],{},[2542,36701,36546],{"mathvariant":2748},[2542,36703,4959],{},[2689,36705,20769],{},[30640,36707,36708,36715],{},[2536,36709,36710,36713],{},[2542,36711,36712],{},"max",[2689,36714,36690],{},[2536,36716,36717,36719,36721,36723,36725,36727,36729],{},[2542,36718,9139],{},[2689,36720,36697],{},[2689,36722,12248],{"stretchy":2755},[2542,36724,49],{},[2689,36726,2819],{"separator":2557},[2542,36728,13102],{},[2689,36730,12273],{"stretchy":2755},[2542,36732,2749],{"mathvariant":2748},[2542,36734,18],{},[2689,36736,580],{"stretchy":2755},[2542,36738,9139],{},[2689,36740,3649],{"stretchy":2755},[2542,36742,2749],{"mathvariant":2748},[2714,36744],{"width":9455},[2549,36746,36747],{"encoding":2551},"\\beta(a, b, d) = \\min_{p \\in \\Pi^*_d} \\max_{x \\in [a, b]} |p(x)| \\quad",[507,36749,36751,36797],{"className":36750,"ariaHidden":2557},[2556],[507,36752,36754,36757,36761,36764,36767,36770,36773,36776,36779,36782,36785,36788,36791,36794],{"className":36753},[2561],[507,36755],{"className":36756,"style":2769},[2565],[507,36758,36665],{"className":36759,"style":36760},[2570,2611],"margin-right:0.0528em;",[507,36762,580],{"className":36763},[2941],[507,36765,49],{"className":36766},[2570,2611],[507,36768,2819],{"className":36769},[2961],[507,36771],{"className":36772,"style":2965},[2714],[507,36774,13102],{"className":36775},[2570,2611],[507,36777,2819],{"className":36778},[2961],[507,36780],{"className":36781,"style":2965},[2714],[507,36783,4959],{"className":36784},[2570,2611],[507,36786,3649],{"className":36787},[2780],[507,36789],{"className":36790,"style":2919},[2714],[507,36792,573],{"className":36793},[2923],[507,36795],{"className":36796,"style":2919},[2714],[507,36798,36800,36804,36912,36915,36982,36985,36988,36991,36994,36997,37000,37003],{"className":36799},[2561],[507,36801],{"className":36802,"style":36803},[2565],"height:1.7374em;vertical-align:-0.9874em;",[507,36805,36807],{"className":36806},[7570,28286],[507,36808,36810,36903],{"className":36809},[2583,3200],[507,36811,36813,36900],{"className":36812},[2587],[507,36814,36817,36890],{"className":36815,"style":36816},[2591],"height:0.6679em;",[507,36818,36820,36823],{"style":36819},"top:-2.3557em;margin-left:0em;",[507,36821],{"className":36822,"style":4310},[2599],[507,36824,36826],{"className":36825},[2604,2605,2606,2607],[507,36827,36829,36832,36835],{"className":36828},[2570,2607],[507,36830,18],{"className":36831},[2570,2611,2607],[507,36833,36697],{"className":36834},[2923,2607],[507,36836,36838,36841],{"className":36837},[2570,2607],[507,36839,36546],{"className":36840},[2570,2607],[507,36842,36844],{"className":36843},[2579],[507,36845,36847,36881],{"className":36846},[2583,3200],[507,36848,36850,36878],{"className":36849},[2587],[507,36851,36854,36866],{"className":36852,"style":36853},[2591],"height:0.6771em;",[507,36855,36857,36860],{"style":36856},"top:-2.1528em;margin-left:0em;margin-right:0.0714em;",[507,36858],{"className":36859,"style":4812},[2599],[507,36861,36863],{"className":36862},[2604,4816,2948,2607],[507,36864,4959],{"className":36865},[2570,2611,2607],[507,36867,36869,36872],{"style":36868},"top:-2.8448em;margin-right:0.0714em;",[507,36870],{"className":36871,"style":4812},[2599],[507,36873,36875],{"className":36874},[2604,4816,2948,2607],[507,36876,20769],{"className":36877},[2719,2607],[507,36879,3225],{"className":36880},[3224],[507,36882,36884],{"className":36883},[2587],[507,36885,36888],{"className":36886,"style":36887},[2591],"height:0.3472em;",[507,36889],{},[507,36891,36892,36895],{"style":4306},[507,36893],{"className":36894,"style":4310},[2599],[507,36896,36897],{},[507,36898,25230],{"className":36899},[7570],[507,36901,3225],{"className":36902},[3224],[507,36904,36906],{"className":36905},[2587],[507,36907,36910],{"className":36908,"style":36909},[2591],"height:0.9874em;",[507,36911],{},[507,36913],{"className":36914,"style":2965},[2714],[507,36916,36918],{"className":36917},[7570,28286],[507,36919,36921,36973],{"className":36920},[2583,3200],[507,36922,36924,36970],{"className":36923},[2587],[507,36925,36927,36960],{"className":36926,"style":2639},[2591],[507,36928,36930,36933],{"style":36929},"top:-2.309em;margin-left:0em;",[507,36931],{"className":36932,"style":4310},[2599],[507,36934,36936],{"className":36935},[2604,2605,2606,2607],[507,36937,36939,36942,36945,36948,36951,36954,36957],{"className":36938},[2570,2607],[507,36940,9139],{"className":36941},[2570,2611,2607],[507,36943,36697],{"className":36944},[2923,2607],[507,36946,12248],{"className":36947},[2941,2607],[507,36949,49],{"className":36950},[2570,2611,2607],[507,36952,2819],{"className":36953},[2961,2607],[507,36955,13102],{"className":36956},[2570,2611,2607],[507,36958,12273],{"className":36959},[2780,2607],[507,36961,36962,36965],{"style":4306},[507,36963],{"className":36964,"style":4310},[2599],[507,36966,36967],{},[507,36968,36712],{"className":36969},[7570],[507,36971,3225],{"className":36972},[3224],[507,36974,36976],{"className":36975},[2587],[507,36977,36980],{"className":36978,"style":36979},[2591],"height:0.966em;",[507,36981],{},[507,36983],{"className":36984,"style":2965},[2714],[507,36986,2749],{"className":36987},[2570],[507,36989,18],{"className":36990},[2570,2611],[507,36992,580],{"className":36993},[2941],[507,36995,9139],{"className":36996},[2570,2611],[507,36998,3649],{"className":36999},[2780],[507,37001,2749],{"className":37002},[2570],[507,37004],{"className":37005,"style":9774},[2714],[18,37007,37008],{},"is",[507,37010,37012],{"className":37011},[2784],[507,37013,37015,37118],{"className":37014},[2523],[507,37016,37018],{"className":37017},[2527],[2529,37019,37020],{"xmlns":2531,"display":2793},[2533,37021,37022,37115],{},[2536,37023,37024,37030,37032,37034,37036,37038,37111,37113],{},[2539,37025,37026,37028],{},[2542,37027,18],{},[2689,37029,20769],{},[2689,37031,580],{"stretchy":2755},[2542,37033,9139],{},[2689,37035,3649],{"stretchy":2755},[2689,37037,573],{},[9491,37039,37040,37079],{},[2536,37041,37042,37049],{},[3168,37043,37044,37047],{},[2542,37045,37046],{},"T",[2542,37048,4959],{},[2536,37050,37051,37053,37077],{},[2689,37052,580],{"fence":2557},[9491,37054,37055,37069],{},[2536,37056,37057,37059,37061,37063,37065,37067],{},[2542,37058,13102],{},[2689,37060,2107],{},[2542,37062,49],{},[2689,37064,2691],{},[2693,37066,584],{},[2542,37068,9139],{},[2536,37070,37071,37073,37075],{},[2542,37072,13102],{},[2689,37074,2691],{},[2542,37076,49],{},[2689,37078,3649],{"fence":2557},[2536,37080,37081,37087],{},[3168,37082,37083,37085],{},[2542,37084,37046],{},[2542,37086,4959],{},[2536,37088,37089,37091,37109],{},[2689,37090,580],{"fence":2557},[9491,37092,37093,37101],{},[2536,37094,37095,37097,37099],{},[2542,37096,13102],{},[2689,37098,2107],{},[2542,37100,49],{},[2536,37102,37103,37105,37107],{},[2542,37104,13102],{},[2689,37106,2691],{},[2542,37108,49],{},[2689,37110,3649],{"fence":2557},[2689,37112,2819],{"separator":2557},[2714,37114],{"width":9455},[2549,37116,37117],{"encoding":2551},"p^*(x) = \\frac{T_d\\left(\\frac{b + a - 2x}{b - a}\\right)}{T_d\\left(\\frac{b + a}{b - a}\\right)}, \\quad",[507,37119,37121,37175],{"className":37120,"ariaHidden":2557},[2556],[507,37122,37124,37127,37157,37160,37163,37166,37169,37172],{"className":37123},[2561],[507,37125],{"className":37126,"style":2769},[2565],[507,37128,37130,37133],{"className":37129},[2570],[507,37131,18],{"className":37132},[2570,2611],[507,37134,37136],{"className":37135},[2579],[507,37137,37139],{"className":37138},[2583],[507,37140,37142],{"className":37141},[2587],[507,37143,37146],{"className":37144,"style":37145},[2591],"height:0.7387em;",[507,37147,37148,37151],{"style":2906},[507,37149],{"className":37150,"style":2600},[2599],[507,37152,37154],{"className":37153},[2604,2605,2606,2607],[507,37155,20769],{"className":37156},[2719,2607],[507,37158,580],{"className":37159},[2941],[507,37161,9139],{"className":37162},[2570,2611],[507,37164,3649],{"className":37165},[2780],[507,37167],{"className":37168,"style":2919},[2714],[507,37170,573],{"className":37171},[2923],[507,37173],{"className":37174,"style":2919},[2714],[507,37176,37178,37182,37531,37534],{"className":37177},[2561],[507,37179],{"className":37180,"style":37181},[2565],"height:2.8469em;vertical-align:-1.1734em;",[507,37183,37185,37188,37528],{"className":37184},[2570],[507,37186],{"className":37187},[2941,9793],[507,37189,37191],{"className":37190},[9491],[507,37192,37194,37519],{"className":37193},[2583,3200],[507,37195,37197,37516],{"className":37196},[2587],[507,37198,37201,37352,37360],{"className":37199,"style":37200},[2591],"height:1.6734em;",[507,37202,37204,37207],{"style":37203},"top:-2.2299em;",[507,37205],{"className":37206,"style":4310},[2599],[507,37208,37210,37250,37253],{"className":37209},[2570],[507,37211,37213,37216],{"className":37212},[2570],[507,37214,37046],{"className":37215,"style":3220},[2570,2611],[507,37217,37219],{"className":37218},[2579],[507,37220,37222,37242],{"className":37221},[2583,3200],[507,37223,37225,37239],{"className":37224},[2587],[507,37226,37228],{"className":37227,"style":11243},[2591],[507,37229,37230,37233],{"style":7637},[507,37231],{"className":37232,"style":2600},[2599],[507,37234,37236],{"className":37235},[2604,2605,2606,2607],[507,37237,4959],{"className":37238},[2570,2611,2607],[507,37240,3225],{"className":37241},[3224],[507,37243,37245],{"className":37244},[2587],[507,37246,37248],{"className":37247,"style":3232},[2591],[507,37249],{},[507,37251],{"className":37252,"style":2965},[2714],[507,37254,37256,37262,37346],{"className":37255},[2937],[507,37257,37259],{"className":37258,"style":2943},[2941,2942],[507,37260,580],{"className":37261},[2947,2948],[507,37263,37265,37268,37343],{"className":37264},[2570],[507,37266],{"className":37267},[2941,9793],[507,37269,37271],{"className":37270},[9491],[507,37272,37274,37334],{"className":37273},[2583,3200],[507,37275,37277,37331],{"className":37276},[2587],[507,37278,37281,37302,37310],{"className":37279,"style":37280},[2591],"height:0.8801em;",[507,37282,37284,37287],{"style":37283},"top:-2.655em;",[507,37285],{"className":37286,"style":4310},[2599],[507,37288,37290],{"className":37289},[2604,2605,2606,2607],[507,37291,37293,37296,37299],{"className":37292},[2570,2607],[507,37294,13102],{"className":37295},[2570,2611,2607],[507,37297,2691],{"className":37298},[2719,2607],[507,37300,49],{"className":37301},[2570,2611,2607],[507,37303,37304,37307],{"style":9878},[507,37305],{"className":37306,"style":4310},[2599],[507,37308],{"className":37309,"style":9886},[9885],[507,37311,37313,37316],{"style":37312},"top:-3.394em;",[507,37314],{"className":37315,"style":4310},[2599],[507,37317,37319],{"className":37318},[2604,2605,2606,2607],[507,37320,37322,37325,37328],{"className":37321},[2570,2607],[507,37323,13102],{"className":37324},[2570,2611,2607],[507,37326,2107],{"className":37327},[2719,2607],[507,37329,49],{"className":37330},[2570,2611,2607],[507,37332,3225],{"className":37333},[3224],[507,37335,37337],{"className":37336},[2587],[507,37338,37341],{"className":37339,"style":37340},[2591],"height:0.4033em;",[507,37342],{},[507,37344],{"className":37345},[2780,9793],[507,37347,37349],{"className":37348,"style":2943},[2780,2942],[507,37350,3649],{"className":37351},[2947,2948],[507,37353,37354,37357],{"style":9878},[507,37355],{"className":37356,"style":4310},[2599],[507,37358],{"className":37359,"style":9886},[9885],[507,37361,37363,37366],{"style":37362},"top:-3.7933em;",[507,37364],{"className":37365,"style":4310},[2599],[507,37367,37369,37409,37412],{"className":37368},[2570],[507,37370,37372,37375],{"className":37371},[2570],[507,37373,37046],{"className":37374,"style":3220},[2570,2611],[507,37376,37378],{"className":37377},[2579],[507,37379,37381,37401],{"className":37380},[2583,3200],[507,37382,37384,37398],{"className":37383},[2587],[507,37385,37387],{"className":37386,"style":11243},[2591],[507,37388,37389,37392],{"style":7637},[507,37390],{"className":37391,"style":2600},[2599],[507,37393,37395],{"className":37394},[2604,2605,2606,2607],[507,37396,4959],{"className":37397},[2570,2611,2607],[507,37399,3225],{"className":37400},[3224],[507,37402,37404],{"className":37403},[2587],[507,37405,37407],{"className":37406,"style":3232},[2591],[507,37408],{},[507,37410],{"className":37411,"style":2965},[2714],[507,37413,37415,37421,37510],{"className":37414},[2937],[507,37416,37418],{"className":37417,"style":2943},[2941,2942],[507,37419,580],{"className":37420},[2947,2948],[507,37422,37424,37427,37507],{"className":37423},[2570],[507,37425],{"className":37426},[2941,9793],[507,37428,37430],{"className":37429},[9491],[507,37431,37433,37499],{"className":37432},[2583,3200],[507,37434,37436,37496],{"className":37435},[2587],[507,37437,37439,37459,37467],{"className":37438,"style":37280},[2591],[507,37440,37441,37444],{"style":37283},[507,37442],{"className":37443,"style":4310},[2599],[507,37445,37447],{"className":37446},[2604,2605,2606,2607],[507,37448,37450,37453,37456],{"className":37449},[2570,2607],[507,37451,13102],{"className":37452},[2570,2611,2607],[507,37454,2691],{"className":37455},[2719,2607],[507,37457,49],{"className":37458},[2570,2611,2607],[507,37460,37461,37464],{"style":9878},[507,37462],{"className":37463,"style":4310},[2599],[507,37465],{"className":37466,"style":9886},[9885],[507,37468,37469,37472],{"style":37312},[507,37470],{"className":37471,"style":4310},[2599],[507,37473,37475],{"className":37474},[2604,2605,2606,2607],[507,37476,37478,37481,37484,37487,37490,37493],{"className":37477},[2570,2607],[507,37479,13102],{"className":37480},[2570,2611,2607],[507,37482,2107],{"className":37483},[2719,2607],[507,37485,49],{"className":37486},[2570,2611,2607],[507,37488,2691],{"className":37489},[2719,2607],[507,37491,584],{"className":37492},[2570,2607],[507,37494,9139],{"className":37495},[2570,2611,2607],[507,37497,3225],{"className":37498},[3224],[507,37500,37502],{"className":37501},[2587],[507,37503,37505],{"className":37504,"style":37340},[2591],[507,37506],{},[507,37508],{"className":37509},[2780,9793],[507,37511,37513],{"className":37512,"style":2943},[2780,2942],[507,37514,3649],{"className":37515},[2947,2948],[507,37517,3225],{"className":37518},[3224],[507,37520,37522],{"className":37521},[2587],[507,37523,37526],{"className":37524,"style":37525},[2591],"height:1.1734em;",[507,37527],{},[507,37529],{"className":37530},[2780,9793],[507,37532,2819],{"className":37533},[2961],[507,37535],{"className":37536,"style":9774},[2714],[18,37538,37539],{},"and the corresponding minimal value is",[507,37541,37543],{"className":37542},[2784],[507,37544,37546,37614],{"className":37545},[2523],[507,37547,37549],{"className":37548},[2527],[2529,37550,37551],{"xmlns":2531,"display":2793},[2533,37552,37553,37611],{},[2536,37554,37555,37557,37559,37561,37563,37565,37567,37569,37571,37573,37585,37609],{},[2542,37556,36665],{},[2689,37558,580],{"stretchy":2755},[2542,37560,49],{},[2689,37562,2819],{"separator":2557},[2542,37564,13102],{},[2689,37566,2819],{"separator":2557},[2542,37568,4959],{},[2689,37570,3649],{"stretchy":2755},[2689,37572,573],{},[3775,37574,37575,37577,37579],{},[2542,37576,37046],{},[2542,37578,4959],{},[2536,37580,37581,37583],{},[2689,37582,2691],{},[2693,37584,625],{},[2536,37586,37587,37589,37607],{},[2689,37588,580],{"fence":2557},[9491,37590,37591,37599],{},[2536,37592,37593,37595,37597],{},[2542,37594,13102],{},[2689,37596,2107],{},[2542,37598,49],{},[2536,37600,37601,37603,37605],{},[2542,37602,13102],{},[2689,37604,2691],{},[2542,37606,49],{},[2689,37608,3649],{"fence":2557},[2542,37610,53],{"mathvariant":2748},[2549,37612,37613],{"encoding":2551},"\\beta(a, b, d) = T_d^{-1}\\left(\\frac{b + a}{b - a}\\right).",[507,37615,37617,37662],{"className":37616,"ariaHidden":2557},[2556],[507,37618,37620,37623,37626,37629,37632,37635,37638,37641,37644,37647,37650,37653,37656,37659],{"className":37619},[2561],[507,37621],{"className":37622,"style":2769},[2565],[507,37624,36665],{"className":37625,"style":36760},[2570,2611],[507,37627,580],{"className":37628},[2941],[507,37630,49],{"className":37631},[2570,2611],[507,37633,2819],{"className":37634},[2961],[507,37636],{"className":37637,"style":2965},[2714],[507,37639,13102],{"className":37640},[2570,2611],[507,37642,2819],{"className":37643},[2961],[507,37645],{"className":37646,"style":2965},[2714],[507,37648,4959],{"className":37649},[2570,2611],[507,37651,3649],{"className":37652},[2780],[507,37654],{"className":37655,"style":2919},[2714],[507,37657,573],{"className":37658},[2923],[507,37660],{"className":37661,"style":2919},[2714],[507,37663,37665,37669,37728,37731,37834,37837],{"className":37664},[2561],[507,37666],{"className":37667,"style":37668},[2565],"height:2.4em;vertical-align:-0.95em;",[507,37670,37672,37675],{"className":37671},[2570],[507,37673,37046],{"className":37674,"style":3220},[2570,2611],[507,37676,37678],{"className":37677},[2579],[507,37679,37681,37719],{"className":37680},[2583,3200],[507,37682,37684,37716],{"className":37683},[2587],[507,37685,37687,37699],{"className":37686,"style":3002},[2591],[507,37688,37690,37693],{"style":37689},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[507,37691],{"className":37692,"style":2600},[2599],[507,37694,37696],{"className":37695},[2604,2605,2606,2607],[507,37697,4959],{"className":37698},[2570,2611,2607],[507,37700,37701,37704],{"style":2906},[507,37702],{"className":37703,"style":2600},[2599],[507,37705,37707],{"className":37706},[2604,2605,2606,2607],[507,37708,37710,37713],{"className":37709},[2570,2607],[507,37711,2691],{"className":37712},[2570,2607],[507,37714,625],{"className":37715},[2570,2607],[507,37717,3225],{"className":37718},[3224],[507,37720,37722],{"className":37721},[2587],[507,37723,37726],{"className":37724,"style":37725},[2591],"height:0.2914em;",[507,37727],{},[507,37729],{"className":37730,"style":2965},[2714],[507,37732,37734,37740,37828],{"className":37733},[2937],[507,37735,37737],{"className":37736,"style":2943},[2941,2942],[507,37738,580],{"className":37739},[2947,2606],[507,37741,37743,37746,37825],{"className":37742},[2570],[507,37744],{"className":37745},[2941,9793],[507,37747,37749],{"className":37748},[9491],[507,37750,37752,37816],{"className":37751},[2583,3200],[507,37753,37755,37813],{"className":37754},[2587],[507,37756,37759,37782,37790],{"className":37757,"style":37758},[2591],"height:1.3714em;",[507,37760,37761,37764],{"style":11717},[507,37762],{"className":37763,"style":4310},[2599],[507,37765,37767,37770,37773,37776,37779],{"className":37766},[2570],[507,37768,13102],{"className":37769},[2570,2611],[507,37771],{"className":37772,"style":2715},[2714],[507,37774,2691],{"className":37775},[2719],[507,37777],{"className":37778,"style":2715},[2714],[507,37780,49],{"className":37781},[2570,2611],[507,37783,37784,37787],{"style":9878},[507,37785],{"className":37786,"style":4310},[2599],[507,37788],{"className":37789,"style":9886},[9885],[507,37791,37792,37795],{"style":9889},[507,37793],{"className":37794,"style":4310},[2599],[507,37796,37798,37801,37804,37807,37810],{"className":37797},[2570],[507,37799,13102],{"className":37800},[2570,2611],[507,37802],{"className":37803,"style":2715},[2714],[507,37805,2107],{"className":37806},[2719],[507,37808],{"className":37809,"style":2715},[2714],[507,37811,49],{"className":37812},[2570,2611],[507,37814,3225],{"className":37815},[3224],[507,37817,37819],{"className":37818},[2587],[507,37820,37823],{"className":37821,"style":37822},[2591],"height:0.7693em;",[507,37824],{},[507,37826],{"className":37827},[2780,9793],[507,37829,37831],{"className":37830,"style":2943},[2780,2942],[507,37832,3649],{"className":37833},[2947,2606],[507,37835],{"className":37836,"style":2965},[2714],[507,37838,53],{"className":37839},[2570],[18,37841,37842,37843,37895],{},"We want to convert this into a function that can be expressed naturally in terms of complex exponentials, because those are the real time-evolutions that generate the quantum Krylov space.\nTo do so, it is convenient to introduce the following transformation of energies within the spectral range of the Hamiltonian to numbers in the range ",[507,37844,37846,37868],{"className":37845},[2523],[507,37847,37849],{"className":37848},[2527],[2529,37850,37851],{"xmlns":2531},[2533,37852,37853,37865],{},[2536,37854,37855,37857,37859,37861,37863],{},[2689,37856,12248],{"stretchy":2755},[2693,37858,601],{},[2689,37860,2819],{"separator":2557},[2693,37862,625],{},[2689,37864,12273],{"stretchy":2755},[2549,37866,37867],{"encoding":2551},"[0,1]",[507,37869,37871],{"className":37870,"ariaHidden":2557},[2556],[507,37872,37874,37877,37880,37883,37886,37889,37892],{"className":37873},[2561],[507,37875],{"className":37876,"style":2769},[2565],[507,37878,12248],{"className":37879},[2941],[507,37881,601],{"className":37882},[2570],[507,37884,2819],{"className":37885},[2961],[507,37887],{"className":37888,"style":2965},[2714],[507,37890,625],{"className":37891},[2570],[507,37893,12273],{"className":37894},[2780],": define",[507,37897,37899],{"className":37898},[2784],[507,37900,37902,37965],{"className":37901},[2523],[507,37903,37905],{"className":37904},[2527],[2529,37906,37907],{"xmlns":2531,"display":2793},[2533,37908,37909,37962],{},[2536,37910,37911,37914,37916,37918,37920,37922,37960],{},[2542,37912,37913],{},"g",[2689,37915,580],{"stretchy":2755},[2542,37917,6182],{},[2689,37919,3649],{"stretchy":2755},[2689,37921,573],{},[9491,37923,37924,37958],{},[2536,37925,37926,37928,37930,37933,37935,37938,37940,37942,37944,37950,37952,37954,37956],{},[2693,37927,625],{},[2689,37929,2691],{},[2542,37931,37932],{},"cos",[2689,37934,36690],{},[2689,37936,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},"1.2em",[2689,37939,580],{"stretchy":2755},[2542,37941,6182],{},[2689,37943,2691],{},[3168,37945,37946,37948],{},[2542,37947,6182],{},[2693,37949,601],{},[2689,37951,3649],{"stretchy":2755},[2542,37953,4959],{},[2542,37955,3298],{},[2689,37957,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2693,37959,584],{},[2689,37961,2819],{"separator":2557},[2549,37963,37964],{"encoding":2551},"g(E) = \\frac{1-\\cos\\big((E-E_0)dt\\big)}{2},",[507,37966,37968,37995],{"className":37967,"ariaHidden":2557},[2556],[507,37969,37971,37974,37977,37980,37983,37986,37989,37992],{"className":37970},[2561],[507,37972],{"className":37973,"style":2769},[2565],[507,37975,37913],{"className":37976,"style":2776},[2570,2611],[507,37978,580],{"className":37979},[2941],[507,37981,6182],{"className":37982,"style":5892},[2570,2611],[507,37984,3649],{"className":37985},[2780],[507,37987],{"className":37988,"style":2919},[2714],[507,37990,573],{"className":37991},[2923],[507,37993],{"className":37994,"style":2919},[2714],[507,37996,37998,38002,38157],{"className":37997},[2561],[507,37999],{"className":38000,"style":38001},[2565],"height:2.276em;vertical-align:-0.686em;",[507,38003,38005,38008,38154],{"className":38004},[2570],[507,38006],{"className":38007},[2941,9793],[507,38009,38011],{"className":38010},[9491],[507,38012,38014,38146],{"className":38013},[2583,3200],[507,38015,38017,38143],{"className":38016},[2587],[507,38018,38021,38032,38040],{"className":38019,"style":38020},[2591],"height:1.59em;",[507,38022,38023,38026],{"style":11717},[507,38024],{"className":38025,"style":4310},[2599],[507,38027,38029],{"className":38028},[2570],[507,38030,584],{"className":38031},[2570],[507,38033,38034,38037],{"style":9878},[507,38035],{"className":38036,"style":4310},[2599],[507,38038],{"className":38039,"style":9886},[9885],[507,38041,38043,38046],{"style":38042},"top:-3.74em;",[507,38044],{"className":38045,"style":4310},[2599],[507,38047,38049,38052,38055,38058,38061,38064,38067,38073,38076,38079,38082,38085,38088,38128,38131,38134,38137],{"className":38048},[2570],[507,38050,625],{"className":38051},[2570],[507,38053],{"className":38054,"style":2715},[2714],[507,38056,2691],{"className":38057},[2719],[507,38059],{"className":38060,"style":2715},[2714],[507,38062,37932],{"className":38063},[7570],[507,38065],{"className":38066,"style":2965},[2714],[507,38068,38070],{"className":38069},[2570],[507,38071,580],{"className":38072},[2947,2948],[507,38074,580],{"className":38075},[2941],[507,38077,6182],{"className":38078,"style":5892},[2570,2611],[507,38080],{"className":38081,"style":2715},[2714],[507,38083,2691],{"className":38084},[2719],[507,38086],{"className":38087,"style":2715},[2714],[507,38089,38091,38094],{"className":38090},[2570],[507,38092,6182],{"className":38093,"style":5892},[2570,2611],[507,38095,38097],{"className":38096},[2579],[507,38098,38100,38120],{"className":38099},[2583,3200],[507,38101,38103,38117],{"className":38102},[2587],[507,38104,38106],{"className":38105,"style":14281},[2591],[507,38107,38108,38111],{"style":6014},[507,38109],{"className":38110,"style":2600},[2599],[507,38112,38114],{"className":38113},[2604,2605,2606,2607],[507,38115,601],{"className":38116},[2570,2607],[507,38118,3225],{"className":38119},[3224],[507,38121,38123],{"className":38122},[2587],[507,38124,38126],{"className":38125,"style":3232},[2591],[507,38127],{},[507,38129,3649],{"className":38130},[2780],[507,38132,4959],{"className":38133},[2570,2611],[507,38135,3298],{"className":38136},[2570,2611],[507,38138,38140],{"className":38139},[2570],[507,38141,3649],{"className":38142},[2947,2948],[507,38144,3225],{"className":38145},[3224],[507,38147,38149],{"className":38148},[2587],[507,38150,38152],{"className":38151,"style":11755},[2591],[507,38153],{},[507,38155],{"className":38156},[2780,9793],[507,38158,2819],{"className":38159},[2961],[18,38161,28454,38162,38195,38196,38402,38403,10799,38510,38554,38555,38583,38584,53],{},[507,38163,38165,38180],{"className":38164},[2523],[507,38166,38168],{"className":38167},[2527],[2529,38169,38170],{"xmlns":2531},[2533,38171,38172,38178],{},[2536,38173,38174,38176],{},[2542,38175,4959],{},[2542,38177,3298],{},[2549,38179,5463],{"encoding":2551},[507,38181,38183],{"className":38182,"ariaHidden":2557},[2556],[507,38184,38186,38189,38192],{"className":38185},[2561],[507,38187],{"className":38188,"style":5434},[2565],[507,38190,4959],{"className":38191},[2570,2611],[507,38193,3298],{"className":38194},[2570,2611]," is a timestep such that ",[507,38197,38199,38243],{"className":38198},[2523],[507,38200,38202],{"className":38201},[2527],[2529,38203,38204],{"xmlns":2531},[2533,38205,38206,38240],{},[2536,38207,38208,38210,38212,38214,38220,38222,38224,38226,38232,38234,38236,38238],{},[2689,38209,2691],{},[2542,38211,8563],{},[2689,38213,5677],{},[3168,38215,38216,38218],{},[2542,38217,6182],{},[2693,38219,601],{},[2542,38221,4959],{},[2542,38223,3298],{},[2689,38225,5677],{},[3168,38227,38228,38230],{},[2542,38229,6182],{},[6167,38231,36712],{},[2542,38233,4959],{},[2542,38235,3298],{},[2689,38237,5677],{},[2542,38239,8563],{},[2549,38241,38242],{"encoding":2551},"-\\pi \u003C E_0dt \u003C E_\\text{max}dt \u003C \\pi",[507,38244,38246,38267,38329,38393],{"className":38245,"ariaHidden":2557},[2556],[507,38247,38249,38252,38255,38258,38261,38264],{"className":38248},[2561],[507,38250],{"className":38251,"style":2707},[2565],[507,38253,2691],{"className":38254},[2570],[507,38256,8563],{"className":38257,"style":2776},[2570,2611],[507,38259],{"className":38260,"style":2919},[2714],[507,38262,5677],{"className":38263},[2923],[507,38265],{"className":38266,"style":2919},[2714],[507,38268,38270,38274,38314,38317,38320,38323,38326],{"className":38269},[2561],[507,38271],{"className":38272,"style":38273},[2565],"height:0.8444em;vertical-align:-0.15em;",[507,38275,38277,38280],{"className":38276},[2570],[507,38278,6182],{"className":38279,"style":5892},[2570,2611],[507,38281,38283],{"className":38282},[2579],[507,38284,38286,38306],{"className":38285},[2583,3200],[507,38287,38289,38303],{"className":38288},[2587],[507,38290,38292],{"className":38291,"style":14281},[2591],[507,38293,38294,38297],{"style":6014},[507,38295],{"className":38296,"style":2600},[2599],[507,38298,38300],{"className":38299},[2604,2605,2606,2607],[507,38301,601],{"className":38302},[2570,2607],[507,38304,3225],{"className":38305},[3224],[507,38307,38309],{"className":38308},[2587],[507,38310,38312],{"className":38311,"style":3232},[2591],[507,38313],{},[507,38315,4959],{"className":38316},[2570,2611],[507,38318,3298],{"className":38319},[2570,2611],[507,38321],{"className":38322,"style":2919},[2714],[507,38324,5677],{"className":38325},[2923],[507,38327],{"className":38328,"style":2919},[2714],[507,38330,38332,38335,38378,38381,38384,38387,38390],{"className":38331},[2561],[507,38333],{"className":38334,"style":38273},[2565],[507,38336,38338,38341],{"className":38337},[2570],[507,38339,6182],{"className":38340,"style":5892},[2570,2611],[507,38342,38344],{"className":38343},[2579],[507,38345,38347,38370],{"className":38346},[2583,3200],[507,38348,38350,38367],{"className":38349},[2587],[507,38351,38353],{"className":38352,"style":4507},[2591],[507,38354,38355,38358],{"style":6014},[507,38356],{"className":38357,"style":2600},[2599],[507,38359,38361],{"className":38360},[2604,2605,2606,2607],[507,38362,38364],{"className":38363},[2570,7039,2607],[507,38365,36712],{"className":38366},[2570,2607],[507,38368,3225],{"className":38369},[3224],[507,38371,38373],{"className":38372},[2587],[507,38374,38376],{"className":38375,"style":3232},[2591],[507,38377],{},[507,38379,4959],{"className":38380},[2570,2611],[507,38382,3298],{"className":38383},[2570,2611],[507,38385],{"className":38386,"style":2919},[2714],[507,38388,5677],{"className":38389},[2923],[507,38391],{"className":38392,"style":2919},[2714],[507,38394,38396,38399],{"className":38395},[2561],[507,38397],{"className":38398,"style":2639},[2565],[507,38400,8563],{"className":38401,"style":2776},[2570,2611],".\nNotice that ",[507,38404,38406,38434],{"className":38405},[2523],[507,38407,38409],{"className":38408},[2527],[2529,38410,38411],{"xmlns":2531},[2533,38412,38413,38431],{},[2536,38414,38415,38417,38419,38425,38427,38429],{},[2542,38416,37913],{},[2689,38418,580],{"stretchy":2755},[3168,38420,38421,38423],{},[2542,38422,6182],{},[2693,38424,601],{},[2689,38426,3649],{"stretchy":2755},[2689,38428,573],{},[2693,38430,601],{},[2549,38432,38433],{"encoding":2551},"g(E_0)=0",[507,38435,38437,38501],{"className":38436,"ariaHidden":2557},[2556],[507,38438,38440,38443,38446,38449,38489,38492,38495,38498],{"className":38439},[2561],[507,38441],{"className":38442,"style":2769},[2565],[507,38444,37913],{"className":38445,"style":2776},[2570,2611],[507,38447,580],{"className":38448},[2941],[507,38450,38452,38455],{"className":38451},[2570],[507,38453,6182],{"className":38454,"style":5892},[2570,2611],[507,38456,38458],{"className":38457},[2579],[507,38459,38461,38481],{"className":38460},[2583,3200],[507,38462,38464,38478],{"className":38463},[2587],[507,38465,38467],{"className":38466,"style":14281},[2591],[507,38468,38469,38472],{"style":6014},[507,38470],{"className":38471,"style":2600},[2599],[507,38473,38475],{"className":38474},[2604,2605,2606,2607],[507,38476,601],{"className":38477},[2570,2607],[507,38479,3225],{"className":38480},[3224],[507,38482,38484],{"className":38483},[2587],[507,38485,38487],{"className":38486,"style":3232},[2591],[507,38488],{},[507,38490,3649],{"className":38491},[2780],[507,38493],{"className":38494,"style":2919},[2714],[507,38496,573],{"className":38497},[2923],[507,38499],{"className":38500,"style":2919},[2714],[507,38502,38504,38507],{"className":38503},[2561],[507,38505],{"className":38506,"style":2729},[2565],[507,38508,601],{"className":38509},[2570],[507,38511,38513,38533],{"className":38512},[2523],[507,38514,38516],{"className":38515},[2527],[2529,38517,38518],{"xmlns":2531},[2533,38519,38520,38530],{},[2536,38521,38522,38524,38526,38528],{},[2542,38523,37913],{},[2689,38525,580],{"stretchy":2755},[2542,38527,6182],{},[2689,38529,3649],{"stretchy":2755},[2549,38531,38532],{"encoding":2551},"g(E)",[507,38534,38536],{"className":38535,"ariaHidden":2557},[2556],[507,38537,38539,38542,38545,38548,38551],{"className":38538},[2561],[507,38540],{"className":38541,"style":2769},[2565],[507,38543,37913],{"className":38544,"style":2776},[2570,2611],[507,38546,580],{"className":38547},[2941],[507,38549,6182],{"className":38550,"style":5892},[2570,2611],[507,38552,3649],{"className":38553},[2780]," grows as ",[507,38556,38558,38571],{"className":38557},[2523],[507,38559,38561],{"className":38560},[2527],[2529,38562,38563],{"xmlns":2531},[2533,38564,38565,38569],{},[2536,38566,38567],{},[2542,38568,6182],{},[2549,38570,6182],{"encoding":2551},[507,38572,38574],{"className":38573,"ariaHidden":2557},[2556],[507,38575,38577,38580],{"className":38576},[2561],[507,38578],{"className":38579,"style":2566},[2565],[507,38581,6182],{"className":38582,"style":5892},[2570,2611]," moves away from ",[507,38585,38587,38604],{"className":38586},[2523],[507,38588,38590],{"className":38589},[2527],[2529,38591,38592],{"xmlns":2531},[2533,38593,38594,38602],{},[2536,38595,38596],{},[3168,38597,38598,38600],{},[2542,38599,6182],{},[2693,38601,601],{},[2549,38603,14253],{"encoding":2551},[507,38605,38607],{"className":38606,"ariaHidden":2557},[2556],[507,38608,38610,38613],{"className":38609},[2561],[507,38611],{"className":38612,"style":3187},[2565],[507,38614,38616,38619],{"className":38615},[2570],[507,38617,6182],{"className":38618,"style":5892},[2570,2611],[507,38620,38622],{"className":38621},[2579],[507,38623,38625,38645],{"className":38624},[2583,3200],[507,38626,38628,38642],{"className":38627},[2587],[507,38629,38631],{"className":38630,"style":14281},[2591],[507,38632,38633,38636],{"style":6014},[507,38634],{"className":38635,"style":2600},[2599],[507,38637,38639],{"className":38638},[2604,2605,2606,2607],[507,38640,601],{"className":38641},[2570,2607],[507,38643,3225],{"className":38644},[3224],[507,38646,38648],{"className":38647},[2587],[507,38649,38651],{"className":38650,"style":3232},[2591],[507,38652],{},[18,38654,38655,38656,38730,38731,622,38860,38911],{},"Now using the polynomial ",[507,38657,38659,38683],{"className":38658},[2523],[507,38660,38662],{"className":38661},[2527],[2529,38663,38664],{"xmlns":2531},[2533,38665,38666,38680],{},[2536,38667,38668,38674,38676,38678],{},[2539,38669,38670,38672],{},[2542,38671,18],{},[2689,38673,20769],{},[2689,38675,580],{"stretchy":2755},[2542,38677,9139],{},[2689,38679,3649],{"stretchy":2755},[2549,38681,38682],{"encoding":2551},"p^*(x)",[507,38684,38686],{"className":38685,"ariaHidden":2557},[2556],[507,38687,38689,38692,38721,38724,38727],{"className":38688},[2561],[507,38690],{"className":38691,"style":2769},[2565],[507,38693,38695,38698],{"className":38694},[2570],[507,38696,18],{"className":38697},[2570,2611],[507,38699,38701],{"className":38700},[2579],[507,38702,38704],{"className":38703},[2583],[507,38705,38707],{"className":38706},[2587],[507,38708,38710],{"className":38709,"style":36582},[2591],[507,38711,38712,38715],{"style":2595},[507,38713],{"className":38714,"style":2600},[2599],[507,38716,38718],{"className":38717},[2604,2605,2606,2607],[507,38719,20769],{"className":38720},[2719,2607],[507,38722,580],{"className":38723},[2941],[507,38725,9139],{"className":38726},[2570,2611],[507,38728,3649],{"className":38729},[2780]," with the parameters a, b, d set to ",[507,38732,38734,38766],{"className":38733},[2523],[507,38735,38737],{"className":38736},[2527],[2529,38738,38739],{"xmlns":2531},[2533,38740,38741,38763],{},[2536,38742,38743,38745,38747,38749,38751,38757,38759,38761],{},[2542,38744,49],{},[2689,38746,573],{},[2542,38748,37913],{},[2689,38750,580],{"stretchy":2755},[3168,38752,38753,38755],{},[2542,38754,6182],{},[2693,38756,601],{},[2689,38758,2107],{},[2542,38760,27524],{},[2689,38762,3649],{"stretchy":2755},[2549,38764,38765],{"encoding":2551},"a = g(E_0 + \\delta)",[507,38767,38769,38787,38848],{"className":38768,"ariaHidden":2557},[2556],[507,38770,38772,38775,38778,38781,38784],{"className":38771},[2561],[507,38773],{"className":38774,"style":2639},[2565],[507,38776,49],{"className":38777},[2570,2611],[507,38779],{"className":38780,"style":2919},[2714],[507,38782,573],{"className":38783},[2923],[507,38785],{"className":38786,"style":2919},[2714],[507,38788,38790,38793,38796,38799,38839,38842,38845],{"className":38789},[2561],[507,38791],{"className":38792,"style":2769},[2565],[507,38794,37913],{"className":38795,"style":2776},[2570,2611],[507,38797,580],{"className":38798},[2941],[507,38800,38802,38805],{"className":38801},[2570],[507,38803,6182],{"className":38804,"style":5892},[2570,2611],[507,38806,38808],{"className":38807},[2579],[507,38809,38811,38831],{"className":38810},[2583,3200],[507,38812,38814,38828],{"className":38813},[2587],[507,38815,38817],{"className":38816,"style":14281},[2591],[507,38818,38819,38822],{"style":6014},[507,38820],{"className":38821,"style":2600},[2599],[507,38823,38825],{"className":38824},[2604,2605,2606,2607],[507,38826,601],{"className":38827},[2570,2607],[507,38829,3225],{"className":38830},[3224],[507,38832,38834],{"className":38833},[2587],[507,38835,38837],{"className":38836,"style":3232},[2591],[507,38838],{},[507,38840],{"className":38841,"style":2715},[2714],[507,38843,2107],{"className":38844},[2719],[507,38846],{"className":38847,"style":2715},[2714],[507,38849,38851,38854,38857],{"className":38850},[2561],[507,38852],{"className":38853,"style":2769},[2565],[507,38855,27524],{"className":38856,"style":27609},[2570,2611],[507,38858,3649],{"className":38859},[2780],[507,38861,38863,38881],{"className":38862},[2523],[507,38864,38866],{"className":38865},[2527],[2529,38867,38868],{"xmlns":2531},[2533,38869,38870,38878],{},[2536,38871,38872,38874,38876],{},[2542,38873,13102],{},[2689,38875,573],{},[2693,38877,625],{},[2549,38879,38880],{"encoding":2551},"b = 1",[507,38882,38884,38902],{"className":38883,"ariaHidden":2557},[2556],[507,38885,38887,38890,38893,38896,38899],{"className":38886},[2561],[507,38888],{"className":38889,"style":5434},[2565],[507,38891,13102],{"className":38892},[2570,2611],[507,38894],{"className":38895,"style":2919},[2714],[507,38897,573],{"className":38898},[2923],[507,38900],{"className":38901,"style":2919},[2714],[507,38903,38905,38908],{"className":38904},[2561],[507,38906],{"className":38907,"style":2729},[2565],[507,38909,625],{"className":38910},[2570],", and d = int(r\u002F2), we define the function:",[507,38913,38915],{"className":38914},[2784],[507,38916,38918,39120],{"className":38917},[2523],[507,38919,38921],{"className":38920},[2527],[2529,38922,38923],{"xmlns":2531,"display":2793},[2533,38924,38925,39117],{},[2536,38926,38927,38929,38931,38933,38935,38937,38943,38957,38959],{},[2542,38928,22278],{},[2689,38930,580],{"stretchy":2755},[2542,38932,6182],{},[2689,38934,3649],{"stretchy":2755},[2689,38936,573],{},[2539,38938,38939,38941],{},[2542,38940,18],{},[2689,38942,20769],{},[2536,38944,38945,38947,38949,38951,38953,38955],{},[2689,38946,580],{"fence":2557},[2542,38948,37913],{},[2689,38950,580],{"stretchy":2755},[2542,38952,6182],{},[2689,38954,3649],{"stretchy":2755},[2689,38956,3649],{"fence":2557},[2689,38958,573],{},[9491,38960,38961,39051],{},[2536,38962,38963,38969],{},[3168,38964,38965,38967],{},[2542,38966,37046],{},[2542,38968,4959],{},[2536,38970,38971,38973,38975,38977,38979,39049],{},[2689,38972,580],{"fence":2557},[2693,38974,625],{},[2689,38976,2107],{},[2693,38978,584],{},[9491,38980,38981,39027],{},[2536,38982,38983,38985,38987,38989,38991,38993,38995,39001,39003,39005,39007,39009,39011,39013,39015,39017,39019,39021,39023,39025],{},[2542,38984,37932],{},[2689,38986,36690],{},[2689,38988,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2689,38990,580],{"stretchy":2755},[2542,38992,6182],{},[2689,38994,2691],{},[3168,38996,38997,38999],{},[2542,38998,6182],{},[2693,39000,601],{},[2689,39002,3649],{"stretchy":2755},[2542,39004,4959],{},[2542,39006,3298],{},[2689,39008,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2689,39010,2691],{},[2542,39012,37932],{},[2689,39014,36690],{},[2689,39016,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2542,39018,27524],{},[6167,39020,25688],{},[2542,39022,4959],{},[2542,39024,3298],{},[2689,39026,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2536,39028,39029,39031,39033,39035,39037,39039,39041,39043,39045,39047],{},[2693,39030,625],{},[2689,39032,2107],{},[2542,39034,37932],{},[2689,39036,36690],{},[2689,39038,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2542,39040,27524],{},[6167,39042,25688],{},[2542,39044,4959],{},[2542,39046,3298],{},[2689,39048,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2689,39050,3649],{"fence":2557},[2536,39052,39053,39059],{},[3168,39054,39055,39057],{},[2542,39056,37046],{},[2542,39058,4959],{},[2536,39060,39061,39063,39065,39067,39069,39115],{},[2689,39062,580],{"fence":2557},[2693,39064,625],{},[2689,39066,2107],{},[2693,39068,584],{},[9491,39070,39071,39093],{},[2536,39072,39073,39075,39077,39079,39081,39083,39085,39087,39089,39091],{},[2693,39074,625],{},[2689,39076,2691],{},[2542,39078,37932],{},[2689,39080,36690],{},[2689,39082,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2542,39084,27524],{},[6167,39086,25688],{},[2542,39088,4959],{},[2542,39090,3298],{},[2689,39092,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2536,39094,39095,39097,39099,39101,39103,39105,39107,39109,39111,39113],{},[2693,39096,625],{},[2689,39098,2107],{},[2542,39100,37932],{},[2689,39102,36690],{},[2689,39104,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2542,39106,27524],{},[6167,39108,25688],{},[2542,39110,4959],{},[2542,39112,3298],{},[2689,39114,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2689,39116,3649],{"fence":2557},[2549,39118,39119],{"encoding":2551},"f(E) = p^* \\left( g(E) \\right) = \\frac{T_d\\left(1 + 2\\frac{\\cos\\big((E-E_0)dt\\big) - \\cos\\big(\\delta\\,dt\\big)}{1 +\\cos\\big(\\delta\\,dt\\big)}\\right)}{T_d\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + 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We can see by inserting 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that ",[507,40167,40169,40188],{"className":40168},[2523],[507,40170,40172],{"className":40171},[2527],[2529,40173,40174],{"xmlns":2531},[2533,40175,40176,40186],{},[2536,40177,40178,40180,40182,40184],{},[2542,40179,22278],{},[2689,40181,580],{"stretchy":2755},[2542,40183,6182],{},[2689,40185,3649],{"stretchy":2755},[2549,40187,27813],{"encoding":2551},[507,40189,40191],{"className":40190,"ariaHidden":2557},[2556],[507,40192,40194,40197,40200,40203,40206],{"className":40193},[2561],[507,40195],{"className":40196,"style":2769},[2565],[507,40198,22278],{"className":40199,"style":27338},[2570,2611],[507,40201,580],{"className":40202},[2941],[507,40204,6182],{"className":40205,"style":5892},[2570,2611],[507,40207,3649],{"className":40208},[2780]," is a trigonometric polynomial of degree ",[507,40211,40213,40226],{"className":40212},[2523],[507,40214,40216],{"className":40215},[2527],[2529,40217,40218],{"xmlns":2531},[2533,40219,40220,40224],{},[2536,40221,40222],{},[2542,40223,4959],{},[2549,40225,4959],{"encoding":2551},[507,40227,40229],{"className":40228,"ariaHidden":2557},[2556],[507,40230,40232,40235],{"className":40231},[2561],[507,40233],{"className":40234,"style":5434},[2565],[507,40236,4959],{"className":40237},[2570,2611],", that is, a linear combination of ",[507,40240,40242,40271],{"className":40241},[2523],[507,40243,40245],{"className":40244},[2527],[2529,40246,40247],{"xmlns":2531},[2533,40248,40249,40269],{},[2536,40250,40251],{},[2539,40252,40253,40255],{},[2542,40254,3286],{},[2536,40256,40257,40259,40261,40263,40265,40267],{},[2542,40258,3293],{},[2542,40260,2372],{},[2542,40262,6182],{},[6167,40264,25688],{},[2542,40266,4959],{},[2542,40268,3298],{},[2549,40270,27868],{"encoding":2551},[507,40272,40274],{"className":40273,"ariaHidden":2557},[2556],[507,40275,40277,40280],{"className":40276},[2561],[507,40278],{"className":40279,"style":3662},[2565],[507,40281,40283,40286],{"className":40282},[2570],[507,40284,3286],{"className":40285},[2570,2611],[507,40287,40289],{"className":40288},[2579],[507,40290,40292],{"className":40291},[2583],[507,40293,40295],{"className":40294},[2587],[507,40296,40298],{"className":40297,"style":3662},[2591],[507,40299,40300,40303],{"style":2595},[507,40301],{"className":40302,"style":2600},[2599],[507,40304,40306],{"className":40305},[2604,2605,2606,2607],[507,40307,40309,40312,40315,40318,40321],{"className":40308},[2570,2607],[507,40310,27910],{"className":40311,"style":6823},[2570,2611,2607],[507,40313,6182],{"className":40314,"style":5892},[2570,2611,2607],[507,40316],{"className":40317,"style":25898},[2714,2607],[507,40319,4959],{"className":40320},[2570,2611,2607],[507,40322,3298],{"className":40323},[2570,2611,2607],[507,40325,40327,40376],{"className":40326},[2523],[507,40328,40330],{"className":40329},[2527],[2529,40331,40332],{"xmlns":2531},[2533,40333,40334,40374],{},[2536,40335,40336,40338,40340,40342,40344,40346,40348,40350,40352,40354,40356,40358,40360,40362,40364,40366,40368,40370,40372],{},[2542,40337,2372],{},[2689,40339,573],{},[2689,40341,2691],{},[2542,40343,4959],{},[2689,40345,2819],{"separator":2557},[2689,40347,2691],{},[2542,40349,4959],{},[2689,40351,2107],{},[2693,40353,625],{},[2689,40355,2819],{"separator":2557},[2542,40357,53],{"mathvariant":2748},[2542,40359,53],{"mathvariant":2748},[2542,40361,53],{"mathvariant":2748},[2689,40363,2819],{"separator":2557},[2542,40365,4959],{},[2689,40367,2691],{},[2693,40369,625],{},[2689,40371,2819],{"separator":2557},[2542,40373,4959],{},[2549,40375,27976],{"encoding":2551},[507,40377,40379,40397,40430,40466],{"className":40378,"ariaHidden":2557},[2556],[507,40380,40382,40385,40388,40391,40394],{"className":40381},[2561],[507,40383],{"className":40384,"style":27986},[2565],[507,40386,2372],{"className":40387,"style":6823},[2570,2611],[507,40389],{"className":40390,"style":2919},[2714],[507,40392,573],{"className":40393},[2923],[507,40395],{"className":40396,"style":2919},[2714],[507,40398,40400,40403,40406,40409,40412,40415,40418,40421,40424,40427],{"className":40399},[2561],[507,40401],{"className":40402,"style":7035},[2565],[507,40404,2691],{"className":40405},[2570],[507,40407,4959],{"className":40408},[2570,2611],[507,40410,2819],{"className":40411},[2961],[507,40413],{"className":40414,"style":2965},[2714],[507,40416,2691],{"className":40417},[2570],[507,40419,4959],{"className":40420},[2570,2611],[507,40422],{"className":40423,"style":2715},[2714],[507,40425,2107],{"className":40426},[2719],[507,40428],{"className":40429,"style":2715},[2714],[507,40431,40433,40436,40439,40442,40445,40448,40451,40454,40457,40460,40463],{"className":40432},[2561],[507,40434],{"className":40435,"style":7035},[2565],[507,40437,625],{"className":40438},[2570],[507,40440,2819],{"className":40441},[2961],[507,40443],{"className":40444,"style":2965},[2714],[507,40446,3032],{"className":40447},[2570],[507,40449,2819],{"className":40450},[2961],[507,40452],{"className":40453,"style":2965},[2714],[507,40455,4959],{"className":40456},[2570,2611],[507,40458],{"className":40459,"style":2715},[2714],[507,40461,2691],{"className":40462},[2719],[507,40464],{"className":40465,"style":2715},[2714],[507,40467,40469,40472,40475,40478,40481],{"className":40468},[2561],[507,40470],{"className":40471,"style":7035},[2565],[507,40473,625],{"className":40474},[2570],[507,40476,2819],{"className":40477},[2961],[507,40479],{"className":40480,"style":2965},[2714],[507,40482,4959],{"className":40483},[2570,2611],". Furthermore, from the definition of ",[507,40486,40488,40511],{"className":40487},[2523],[507,40489,40491],{"className":40490},[2527],[2529,40492,40493],{"xmlns":2531},[2533,40494,40495,40509],{},[2536,40496,40497,40503,40505,40507],{},[2539,40498,40499,40501],{},[2542,40500,18],{},[2689,40502,20769],{},[2689,40504,580],{"stretchy":2755},[2542,40506,9139],{},[2689,40508,3649],{"stretchy":2755},[2549,40510,38682],{"encoding":2551},[507,40512,40514],{"className":40513,"ariaHidden":2557},[2556],[507,40515,40517,40520,40549,40552,40555],{"className":40516},[2561],[507,40518],{"className":40519,"style":2769},[2565],[507,40521,40523,40526],{"className":40522},[2570],[507,40524,18],{"className":40525},[2570,2611],[507,40527,40529],{"className":40528},[2579],[507,40530,40532],{"className":40531},[2583],[507,40533,40535],{"className":40534},[2587],[507,40536,40538],{"className":40537,"style":36582},[2591],[507,40539,40540,40543],{"style":2595},[507,40541],{"className":40542,"style":2600},[2599],[507,40544,40546],{"className":40545},[2604,2605,2606,2607],[507,40547,20769],{"className":40548},[2719,2607],[507,40550,580],{"className":40551},[2941],[507,40553,9139],{"className":40554},[2570,2611],[507,40556,3649],{"className":40557},[2780]," above we have that ",[507,40560,40562,40600],{"className":40561},[2523],[507,40563,40565],{"className":40564},[2527],[2529,40566,40567],{"xmlns":2531},[2533,40568,40569,40597],{},[2536,40570,40571,40573,40575,40581,40583,40585,40587,40589,40591,40593,40595],{},[2542,40572,22278],{},[2689,40574,580],{"stretchy":2755},[3168,40576,40577,40579],{},[2542,40578,6182],{},[2693,40580,601],{},[2689,40582,3649],{"stretchy":2755},[2689,40584,573],{},[2542,40586,18],{},[2689,40588,580],{"stretchy":2755},[2693,40590,601],{},[2689,40592,3649],{"stretchy":2755},[2689,40594,573],{},[2693,40596,625],{},[2549,40598,40599],{"encoding":2551},"f(E_0)=p(0)=1",[507,40601,40603,40667,40694],{"className":40602,"ariaHidden":2557},[2556],[507,40604,40606,40609,40612,40615,40655,40658,40661,40664],{"className":40605},[2561],[507,40607],{"className":40608,"style":2769},[2565],[507,40610,22278],{"className":40611,"style":27338},[2570,2611],[507,40613,580],{"className":40614},[2941],[507,40616,40618,40621],{"className":40617},[2570],[507,40619,6182],{"className":40620,"style":5892},[2570,2611],[507,40622,40624],{"className":40623},[2579],[507,40625,40627,40647],{"className":40626},[2583,3200],[507,40628,40630,40644],{"className":40629},[2587],[507,40631,40633],{"className":40632,"style":14281},[2591],[507,40634,40635,40638],{"style":6014},[507,40636],{"className":40637,"style":2600},[2599],[507,40639,40641],{"className":40640},[2604,2605,2606,2607],[507,40642,601],{"className":40643},[2570,2607],[507,40645,3225],{"className":40646},[3224],[507,40648,40650],{"className":40649},[2587],[507,40651,40653],{"className":40652,"style":3232},[2591],[507,40654],{},[507,40656,3649],{"className":40657},[2780],[507,40659],{"className":40660,"style":2919},[2714],[507,40662,573],{"className":40663},[2923],[507,40665],{"className":40666,"style":2919},[2714],[507,40668,40670,40673,40676,40679,40682,40685,40688,40691],{"className":40669},[2561],[507,40671],{"className":40672,"style":2769},[2565],[507,40674,18],{"className":40675},[2570,2611],[507,40677,580],{"className":40678},[2941],[507,40680,601],{"className":40681},[2570],[507,40683,3649],{"className":40684},[2780],[507,40686],{"className":40687,"style":2919},[2714],[507,40689,573],{"className":40690},[2923],[507,40692],{"className":40693,"style":2919},[2714],[507,40695,40697,40700],{"className":40696},[2561],[507,40698],{"className":40699,"style":2729},[2565],[507,40701,625],{"className":40702},[2570]," and for any ",[507,40705,40707,40720],{"className":40706},[2523],[507,40708,40710],{"className":40709},[2527],[2529,40711,40712],{"xmlns":2531},[2533,40713,40714,40718],{},[2536,40715,40716],{},[2542,40717,6182],{},[2549,40719,6182],{"encoding":2551},[507,40721,40723],{"className":40722,"ariaHidden":2557},[2556],[507,40724,40726,40729],{"className":40725},[2561],[507,40727],{"className":40728,"style":2566},[2565],[507,40730,6182],{"className":40731,"style":5892},[2570,2611]," in the spectral range such that ",[507,40734,40736,40766],{"className":40735},[2523],[507,40737,40739],{"className":40738},[2527],[2529,40740,40741],{"xmlns":2531},[2533,40742,40743,40763],{},[2536,40744,40745,40747,40749,40751,40757,40759,40761],{},[2542,40746,2749],{"mathvariant":2748},[2542,40748,6182],{},[2689,40750,2691],{},[3168,40752,40753,40755],{},[2542,40754,6182],{},[2693,40756,601],{},[2542,40758,2749],{"mathvariant":2748},[2689,40760,1651],{},[2542,40762,27524],{},[2549,40764,40765],{"encoding":2551},"\\vert E-E_0 \\vert > \\delta",[507,40767,40769,40790,40848],{"className":40768,"ariaHidden":2557},[2556],[507,40770,40772,40775,40778,40781,40784,40787],{"className":40771},[2561],[507,40773],{"className":40774,"style":2769},[2565],[507,40776,2749],{"className":40777},[2570],[507,40779,6182],{"className":40780,"style":5892},[2570,2611],[507,40782],{"className":40783,"style":2715},[2714],[507,40785,2691],{"className":40786},[2719],[507,40788],{"className":40789,"style":2715},[2714],[507,40791,40793,40796,40836,40839,40842,40845],{"className":40792},[2561],[507,40794],{"className":40795,"style":2769},[2565],[507,40797,40799,40802],{"className":40798},[2570],[507,40800,6182],{"className":40801,"style":5892},[2570,2611],[507,40803,40805],{"className":40804},[2579],[507,40806,40808,40828],{"className":40807},[2583,3200],[507,40809,40811,40825],{"className":40810},[2587],[507,40812,40814],{"className":40813,"style":14281},[2591],[507,40815,40816,40819],{"style":6014},[507,40817],{"className":40818,"style":2600},[2599],[507,40820,40822],{"className":40821},[2604,2605,2606,2607],[507,40823,601],{"className":40824},[2570,2607],[507,40826,3225],{"className":40827},[3224],[507,40829,40831],{"className":40830},[2587],[507,40832,40834],{"className":40833,"style":3232},[2591],[507,40835],{},[507,40837,2749],{"className":40838},[2570],[507,40840],{"className":40841,"style":2919},[2714],[507,40843,1651],{"className":40844},[2923],[507,40846],{"className":40847,"style":2919},[2714],[507,40849,40851,40854],{"className":40850},[2561],[507,40852],{"className":40853,"style":5434},[2565],[507,40855,27524],{"className":40856,"style":27609},[2570,2611]," we have",[507,40859,40861],{"className":40860},[2784],[507,40862,40864,40978],{"className":40863},[2523],[507,40865,40867],{"className":40866},[2527],[2529,40868,40869],{"xmlns":2531,"display":2793},[2533,40870,40871,40975],{},[2536,40872,40873,40875,40877,40879,40881,40883,40885,40887,40889,40891,40893,40895,40897,40899,40901,40903,40905,40917],{},[2542,40874,2749],{"mathvariant":2748},[2542,40876,22278],{},[2689,40878,580],{"stretchy":2755},[2542,40880,6182],{},[2689,40882,3649],{"stretchy":2755},[2542,40884,2749],{"mathvariant":2748},[2689,40886,27509],{},[2542,40888,36665],{},[2689,40890,580],{"stretchy":2755},[2542,40892,49],{},[2689,40894,2819],{"separator":2557},[2542,40896,13102],{},[2689,40898,2819],{"separator":2557},[2542,40900,4959],{},[2689,40902,3649],{"stretchy":2755},[2689,40904,573],{},[3775,40906,40907,40909,40911],{},[2542,40908,37046],{},[2542,40910,4959],{},[2536,40912,40913,40915],{},[2689,40914,2691],{},[2693,40916,625],{},[2536,40918,40919,40921,40923,40925,40927,40973],{},[2689,40920,580],{"fence":2557},[2693,40922,625],{},[2689,40924,2107],{},[2693,40926,584],{},[9491,40928,40929,40951],{},[2536,40930,40931,40933,40935,40937,40939,40941,40943,40945,40947,40949],{},[2693,40932,625],{},[2689,40934,2691],{},[2542,40936,37932],{},[2689,40938,36690],{},[2689,40940,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2542,40942,27524],{},[6167,40944,25688],{},[2542,40946,4959],{},[2542,40948,3298],{},[2689,40950,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2536,40952,40953,40955,40957,40959,40961,40963,40965,40967,40969,40971],{},[2693,40954,625],{},[2689,40956,2107],{},[2542,40958,37932],{},[2689,40960,36690],{},[2689,40962,580],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2542,40964,27524],{},[6167,40966,25688],{},[2542,40968,4959],{},[2542,40970,3298],{},[2689,40972,3649],{"fence":2755,"stretchy":2557,"minsize":37937,"maxsize":37937},[2689,40974,3649],{"fence":2557},[2549,40976,40977],{"encoding":2551},"|f(E)| \\le \\beta(a, b, d) = T_d^{-1}\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + 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2\\left(1 + \\delta\\right)^{-d} = 2\\left(1 + \\delta\\right)^{-\\lfloor 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N. Yoshioka, M. Amico, W. Kirby et al. \"Diagonalization of large many-body Hamiltonians on a quantum processor\". ",[49,41563,41565],{"href":41564},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.14431","arXiv:2407.14431",[18,41567,41568,41570],{},[507,41569,584],{}," Ethan N. Epperly, Lin Lin, and Yuji Nakatsukasa. \"A theory of quantum subspace diagonalization\". SIAM Journal on Matrix Analysis and Applications 43, 1263–1290 (2022).",[18,41572,41573,41575],{},[507,41574,8226],{}," Å. Björck. \"Numerical methods in matrix computations\". Texts in Applied Mathematics. Springer International Publishing. (2014).",[18,41577,41578,41580],{},[507,41579,12152],{}," William Kirby. \"Analysis of quantum Krylov algorithms with errors\". Quantum 8, 1457 (2024).",[13,41582,41584],{"id":41583},"tutorial-survey","Tutorial survey",[18,41586,41587],{},"Please take this short survey to provide feedback on this tutorial. Your insights will help us improve our content offerings and user experience.",[18,41589,41590],{},[49,41591,41593],{"href":41592},"https:\u002F\u002Fyour.feedback.ibm.com\u002Fjfe\u002Fform\u002FSV_82nennpKIjjD8rQ","Link to survey",[13,41595,940],{"id":939},[18,41597,41598,41599,53],{},"Get ",[49,41600,1037],{"href":1036},[953,41602,41603],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}",{"title":104,"searchDepth":105,"depth":105,"links":41605},[41606,41607,41608,41609,41613,41620,41625,41628,41629,41630,41631,41632],{"id":1061,"depth":105,"text":1062},{"id":1096,"depth":105,"text":1097},{"id":1119,"depth":105,"text":1120},{"id":2510,"depth":105,"text":2511,"children":41610},[41611,41612],{"id":2515,"depth":540,"text":2516},{"id":4084,"depth":540,"text":4085},{"id":14215,"depth":105,"text":14216,"children":41614},[41615,41616,41617,41618],{"id":14219,"depth":540,"text":14220},{"id":20349,"depth":540,"text":20350},{"id":21336,"depth":540,"text":21337},{"id":21433,"depth":540,"text":41619},"Template circuits for calculating matrix elements of S~\\tilde{S}S~ and H~\\tilde{H}H~ via Hadamard test",{"id":21911,"depth":105,"text":21912,"children":41621},[41622,41623,41624],{"id":22025,"depth":540,"text":22026},{"id":22645,"depth":540,"text":22646},{"id":23117,"depth":540,"text":23118},{"id":23977,"depth":105,"text":23978,"children":41626},[41627],{"id":24006,"depth":540,"text":24007},{"id":25618,"depth":105,"text":25619},{"id":36447,"depth":105,"text":36448},{"id":41555,"depth":105,"text":41556},{"id":41583,"depth":105,"text":41584},{"id":939,"depth":105,"text":940},[112,969,970,41634],"Krylov diagonalization",[41636],{"username":1025,"name":41637,"role":41638,"bio":41639,"links":41640},"Mirko Amico","Researcher, Extropic · formerly IBM Research","A researcher at Extropic, previously at IBM Research, and a Qiskit Pulse expert.",[41641],{"label":41642,"href":1036},"Original tutorial ↗",{"username":1025,"name":41637,"role":41644},"Researcher, Extropic","Implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns.","Deep dive · Algorithms",{},"\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization","15 min read",[],{"title":1023,"description":41645},"blog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization",[41654,41655],"algorithms","tutorial","h_4WJ9crI4tBC8SOea5gUgWXCAFpFws4fEwgpxBRWsU",{"id":41658,"title":41659,"authors":41660,"body":41661,"breadcrumb":67655,"builders":67657,"byline":67662,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":116,"description":67663,"draft":125,"extension":126,"eyebrow":67664,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":988,"lessonCount":116,"meta":67665,"navigation":133,"newsItems":116,"next":116,"ogImage":67666,"order":116,"outcomes":116,"path":67667,"publishDate":992,"readingTime":993,"related":67668,"relatedProjects":116,"seo":67669,"stem":67670,"tags":67671,"track":116,"trackName":116,"__hash__":67673},"blog\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization.md","Rabi Oscillations: Visualizing Excited-State 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original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[18,41673,41674],{},"This notebook provides a primer on the quantum dynamics of a driven two-level system by visualizing the excited-state probability under the rotating-wave approximation (RWA). The model treats a qubit as an ideal two-state system driven by a classical microwave tone. This formulation maps the probability of measuring the excited state as a function of drive frequency and pulse duration. The central computational object is the probability surface",[507,41676,41678],{"className":41677},[2784],[507,41679,41681,41712],{"className":41680},[2523],[507,41682,41684],{"className":41683},[2527],[2529,41685,41686],{"xmlns":2531,"display":2793},[2533,41687,41688,41709],{},[2536,41689,41690,41696,41698,41700,41702,41705,41707],{},[3168,41691,41692,41694],{},[2542,41693,3174],{},[2542,41695,3286],{},[2689,41697,580],{"stretchy":2755},[2542,41699,22278],{},[2689,41701,2819],{"separator":2557},[2542,41703,41704],{},"τ",[2689,41706,3649],{"stretchy":2755},[2689,41708,2819],{"separator":2557},[2549,41710,41711],{"encoding":2551},"P_e(f,\\tau),",[507,41713,41715],{"className":41714,"ariaHidden":2557},[2556],[507,41716,41718,41721,41761,41764,41767,41770,41773,41777,41780],{"className":41717},[2561],[507,41719],{"className":41720,"style":2769},[2565],[507,41722,41724,41727],{"className":41723},[2570],[507,41725,3174],{"className":41726,"style":3220},[2570,2611],[507,41728,41730],{"className":41729},[2579],[507,41731,41733,41753],{"className":41732},[2583,3200],[507,41734,41736,41750],{"className":41735},[2587],[507,41737,41739],{"className":41738,"style":4507},[2591],[507,41740,41741,41744],{"style":7637},[507,41742],{"className":41743,"style":2600},[2599],[507,41745,41747],{"className":41746},[2604,2605,2606,2607],[507,41748,3286],{"className":41749},[2570,2611,2607],[507,41751,3225],{"className":41752},[3224],[507,41754,41756],{"className":41755},[2587],[507,41757,41759],{"className":41758,"style":3232},[2591],[507,41760],{},[507,41762,580],{"className":41763},[2941],[507,41765,22278],{"className":41766,"style":27338},[2570,2611],[507,41768,2819],{"className":41769},[2961],[507,41771],{"className":41772,"style":2965},[2714],[507,41774,41704],{"className":41775,"style":41776},[2570,2611],"margin-right:0.1132em;",[507,41778,3649],{"className":41779},[2780],[507,41781,2819],{"className":41782},[2961],[18,41784,28454,41785,41813,41814,41843],{},[507,41786,41788,41801],{"className":41787},[2523],[507,41789,41791],{"className":41790},[2527],[2529,41792,41793],{"xmlns":2531},[2533,41794,41795,41799],{},[2536,41796,41797],{},[2542,41798,22278],{},[2549,41800,22278],{"encoding":2551},[507,41802,41804],{"className":41803,"ariaHidden":2557},[2556],[507,41805,41807,41810],{"className":41806},[2561],[507,41808],{"className":41809,"style":7035},[2565],[507,41811,22278],{"className":41812,"style":27338},[2570,2611]," denotes the applied drive frequency and ",[507,41815,41817,41831],{"className":41816},[2523],[507,41818,41820],{"className":41819},[2527],[2529,41821,41822],{"xmlns":2531},[2533,41823,41824,41828],{},[2536,41825,41826],{},[2542,41827,41704],{},[2549,41829,41830],{"encoding":2551},"\\tau",[507,41832,41834],{"className":41833,"ariaHidden":2557},[2556],[507,41835,41837,41840],{"className":41836},[2561],[507,41838],{"className":41839,"style":2639},[2565],[507,41841,41704],{"className":41842,"style":41776},[2570,2611]," denotes the microwave pulse duration. The notebook renders this probability surface with two-dimensional heatmaps, three-dimensional surfaces, one-dimensional cross sections, and Fast Fourier transform (FFT) spectrograms. Fourier analysis exposes the generalized Rabi frequency as a theoretical ridge in the frequency domain, which connects the time-domain quantum oscillation to its spectral structure.",[13,41845,41847],{"id":41846},"nomenclature-and-definitions","Nomenclature and Definitions",[18,41849,41850],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[41852,41853,41854,41867],"table",{},[41855,41856,41857],"thead",{},[41858,41859,41860,41864],"tr",{},[41861,41862,41863],"th",{},"Symbol or Term",[41861,41865,41866],{},"Meaning",[41868,41869,41870,41917,41963,42009,42136,42213,42291,42326,42547,42644,42681,42727,42878,42955,43180,43215,43308,43316],"tbody",{},[41858,41871,41872,41914],{},[41873,41874,41875],"td",{},[507,41876,41878,41896],{"className":41877},[2523],[507,41879,41881],{"className":41880},[2527],[2529,41882,41883],{"xmlns":2531},[2533,41884,41885,41893],{},[2536,41886,41887,41889,41891],{},[2689,41888,2749],{"stretchy":2755},[2542,41890,37913],{},[2689,41892,2756],{"stretchy":2755},[2549,41894,41895],{"encoding":2551},"\\lvert g\\rangle",[507,41897,41899],{"className":41898,"ariaHidden":2557},[2556],[507,41900,41902,41905,41908,41911],{"className":41901},[2561],[507,41903],{"className":41904,"style":2769},[2565],[507,41906,2749],{"className":41907},[2941],[507,41909,37913],{"className":41910,"style":2776},[2570,2611],[507,41912,2756],{"className":41913},[2780],[41873,41915,41916],{},"Ground state of the qubit",[41858,41918,41919,41960],{},[41873,41920,41921],{},[507,41922,41924,41942],{"className":41923},[2523],[507,41925,41927],{"className":41926},[2527],[2529,41928,41929],{"xmlns":2531},[2533,41930,41931,41939],{},[2536,41932,41933,41935,41937],{},[2689,41934,2749],{"stretchy":2755},[2542,41936,3286],{},[2689,41938,2756],{"stretchy":2755},[2549,41940,41941],{"encoding":2551},"\\lvert e\\rangle",[507,41943,41945],{"className":41944,"ariaHidden":2557},[2556],[507,41946,41948,41951,41954,41957],{"className":41947},[2561],[507,41949],{"className":41950,"style":2769},[2565],[507,41952,2749],{"className":41953},[2941],[507,41955,3286],{"className":41956},[2570,2611],[507,41958,2756],{"className":41959},[2780],[41873,41961,41962],{},"Excited state of the qubit",[41858,41964,41965,42006],{},[41873,41966,41967],{},[507,41968,41970,41988],{"className":41969},[2523],[507,41971,41973],{"className":41972},[2527],[2529,41974,41975],{"xmlns":2531},[2533,41976,41977,41985],{},[2536,41978,41979,41981,41983],{},[2689,41980,2749],{"stretchy":2755},[2542,41982,3793],{},[2689,41984,2756],{"stretchy":2755},[2549,41986,41987],{"encoding":2551},"\\lvert \\psi\\rangle",[507,41989,41991],{"className":41990,"ariaHidden":2557},[2556],[507,41992,41994,41997,42000,42003],{"className":41993},[2561],[507,41995],{"className":41996,"style":2769},[2565],[507,41998,2749],{"className":41999},[2941],[507,42001,3793],{"className":42002,"style":2776},[2570,2611],[507,42004,2756],{"className":42005},[2780],[41873,42007,42008],{},"Quantum state vector in a two-dimensional Hilbert space",[41858,42010,42011,42057],{},[41873,42012,42013],{},[507,42014,42016,42035],{"className":42015},[2523],[507,42017,42019],{"className":42018},[2527],[2529,42020,42021],{"xmlns":2531},[2533,42022,42023,42032],{},[2536,42024,42025,42028,42030],{},[2542,42026,42027],{},"α",[2689,42029,2819],{"separator":2557},[2542,42031,36665],{},[2549,42033,42034],{"encoding":2551},"\\alpha,\\beta",[507,42036,42038],{"className":42037,"ariaHidden":2557},[2556],[507,42039,42041,42044,42048,42051,42054],{"className":42040},[2561],[507,42042],{"className":42043,"style":7035},[2565],[507,42045,42027],{"className":42046,"style":42047},[2570,2611],"margin-right:0.0037em;",[507,42049,2819],{"className":42050},[2961],[507,42052],{"className":42053,"style":2965},[2714],[507,42055,36665],{"className":42056,"style":36760},[2570,2611],[41873,42058,42059,42060,10799,42098],{},"Complex probability amplitudes for ",[507,42061,42063,42080],{"className":42062},[2523],[507,42064,42066],{"className":42065},[2527],[2529,42067,42068],{"xmlns":2531},[2533,42069,42070,42078],{},[2536,42071,42072,42074,42076],{},[2689,42073,2749],{"stretchy":2755},[2542,42075,37913],{},[2689,42077,2756],{"stretchy":2755},[2549,42079,41895],{"encoding":2551},[507,42081,42083],{"className":42082,"ariaHidden":2557},[2556],[507,42084,42086,42089,42092,42095],{"className":42085},[2561],[507,42087],{"className":42088,"style":2769},[2565],[507,42090,2749],{"className":42091},[2941],[507,42093,37913],{"className":42094,"style":2776},[2570,2611],[507,42096,2756],{"className":42097},[2780],[507,42099,42101,42118],{"className":42100},[2523],[507,42102,42104],{"className":42103},[2527],[2529,42105,42106],{"xmlns":2531},[2533,42107,42108,42116],{},[2536,42109,42110,42112,42114],{},[2689,42111,2749],{"stretchy":2755},[2542,42113,3286],{},[2689,42115,2756],{"stretchy":2755},[2549,42117,41941],{"encoding":2551},[507,42119,42121],{"className":42120,"ariaHidden":2557},[2556],[507,42122,42124,42127,42130,42133],{"className":42123},[2561],[507,42125],{"className":42126,"style":2769},[2565],[507,42128,2749],{"className":42129},[2941],[507,42131,3286],{"className":42132},[2570,2611],[507,42134,2756],{"className":42135},[2780],[41858,42137,42138,42210],{},[41873,42139,42140],{},[507,42141,42143,42161],{"className":42142},[2523],[507,42144,42146],{"className":42145},[2527],[2529,42147,42148],{"xmlns":2531},[2533,42149,42150,42158],{},[2536,42151,42152],{},[3168,42153,42154,42156],{},[2542,42155,3174],{},[2542,42157,3286],{},[2549,42159,42160],{"encoding":2551},"P_e",[507,42162,42164],{"className":42163,"ariaHidden":2557},[2556],[507,42165,42167,42170],{"className":42166},[2561],[507,42168],{"className":42169,"style":3187},[2565],[507,42171,42173,42176],{"className":42172},[2570],[507,42174,3174],{"className":42175,"style":3220},[2570,2611],[507,42177,42179],{"className":42178},[2579],[507,42180,42182,42202],{"className":42181},[2583,3200],[507,42183,42185,42199],{"className":42184},[2587],[507,42186,42188],{"className":42187,"style":4507},[2591],[507,42189,42190,42193],{"style":7637},[507,42191],{"className":42192,"style":2600},[2599],[507,42194,42196],{"className":42195},[2604,2605,2606,2607],[507,42197,3286],{"className":42198},[2570,2611,2607],[507,42200,3225],{"className":42201},[3224],[507,42203,42205],{"className":42204},[2587],[507,42206,42208],{"className":42207,"style":3232},[2591],[507,42209],{},[41873,42211,42212],{},"Probability of measuring the qubit in the excited state",[41858,42214,42215,42288],{},[41873,42216,42217],{},[507,42218,42220,42238],{"className":42219},[2523],[507,42221,42223],{"className":42222},[2527],[2529,42224,42225],{"xmlns":2531},[2533,42226,42227,42235],{},[2536,42228,42229],{},[3168,42230,42231,42233],{},[2542,42232,22278],{},[2693,42234,601],{},[2549,42236,42237],{"encoding":2551},"f_0",[507,42239,42241],{"className":42240,"ariaHidden":2557},[2556],[507,42242,42244,42247],{"className":42243},[2561],[507,42245],{"className":42246,"style":7035},[2565],[507,42248,42250,42253],{"className":42249},[2570],[507,42251,22278],{"className":42252,"style":27338},[2570,2611],[507,42254,42256],{"className":42255},[2579],[507,42257,42259,42280],{"className":42258},[2583,3200],[507,42260,42262,42277],{"className":42261},[2587],[507,42263,42265],{"className":42264,"style":14281},[2591],[507,42266,42268,42271],{"style":42267},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[507,42269],{"className":42270,"style":2600},[2599],[507,42272,42274],{"className":42273},[2604,2605,2606,2607],[507,42275,601],{"className":42276},[2570,2607],[507,42278,3225],{"className":42279},[3224],[507,42281,42283],{"className":42282},[2587],[507,42284,42286],{"className":42285,"style":3232},[2591],[507,42287],{},[41873,42289,42290],{},"Natural transition frequency of the qubit",[41858,42292,42293,42323],{},[41873,42294,42295],{},[507,42296,42298,42311],{"className":42297},[2523],[507,42299,42301],{"className":42300},[2527],[2529,42302,42303],{"xmlns":2531},[2533,42304,42305,42309],{},[2536,42306,42307],{},[2542,42308,22278],{},[2549,42310,22278],{"encoding":2551},[507,42312,42314],{"className":42313,"ariaHidden":2557},[2556],[507,42315,42317,42320],{"className":42316},[2561],[507,42318],{"className":42319,"style":7035},[2565],[507,42321,22278],{"className":42322,"style":27338},[2570,2611],[41873,42324,42325],{},"Applied drive frequency",[41858,42327,42328,42401],{},[41873,42329,42330],{},[507,42331,42333,42352],{"className":42332},[2523],[507,42334,42336],{"className":42335},[2527],[2529,42337,42338],{"xmlns":2531},[2533,42339,42340,42349],{},[2536,42341,42342],{},[3168,42343,42344,42347],{},[2542,42345,42346],{},"ω",[2693,42348,601],{},[2549,42350,42351],{"encoding":2551},"\\omega_0",[507,42353,42355],{"className":42354,"ariaHidden":2557},[2556],[507,42356,42358,42361],{"className":42357},[2561],[507,42359],{"className":42360,"style":25197},[2565],[507,42362,42364,42367],{"className":42363},[2570],[507,42365,42346],{"className":42366,"style":2776},[2570,2611],[507,42368,42370],{"className":42369},[2579],[507,42371,42373,42393],{"className":42372},[2583,3200],[507,42374,42376,42390],{"className":42375},[2587],[507,42377,42379],{"className":42378,"style":14281},[2591],[507,42380,42381,42384],{"style":4572},[507,42382],{"className":42383,"style":2600},[2599],[507,42385,42387],{"className":42386},[2604,2605,2606,2607],[507,42388,601],{"className":42389},[2570,2607],[507,42391,3225],{"className":42392},[3224],[507,42394,42396],{"className":42395},[2587],[507,42397,42399],{"className":42398,"style":3232},[2591],[507,42400],{},[41873,42402,42403,42404],{},"Angular transition frequency, ",[507,42405,42407,42437],{"className":42406},[2523],[507,42408,42410],{"className":42409},[2527],[2529,42411,42412],{"xmlns":2531},[2533,42413,42414,42434],{},[2536,42415,42416,42422,42424,42426,42428],{},[3168,42417,42418,42420],{},[2542,42419,42346],{},[2693,42421,601],{},[2689,42423,573],{},[2693,42425,584],{},[2542,42427,8563],{},[3168,42429,42430,42432],{},[2542,42431,22278],{},[2693,42433,601],{},[2549,42435,42436],{"encoding":2551},"\\omega_0=2\\pi f_0",[507,42438,42440,42495],{"className":42439,"ariaHidden":2557},[2556],[507,42441,42443,42446,42486,42489,42492],{"className":42442},[2561],[507,42444],{"className":42445,"style":25197},[2565],[507,42447,42449,42452],{"className":42448},[2570],[507,42450,42346],{"className":42451,"style":2776},[2570,2611],[507,42453,42455],{"className":42454},[2579],[507,42456,42458,42478],{"className":42457},[2583,3200],[507,42459,42461,42475],{"className":42460},[2587],[507,42462,42464],{"className":42463,"style":14281},[2591],[507,42465,42466,42469],{"style":4572},[507,42467],{"className":42468,"style":2600},[2599],[507,42470,42472],{"className":42471},[2604,2605,2606,2607],[507,42473,601],{"className":42474},[2570,2607],[507,42476,3225],{"className":42477},[3224],[507,42479,42481],{"className":42480},[2587],[507,42482,42484],{"className":42483,"style":3232},[2591],[507,42485],{},[507,42487],{"className":42488,"style":2919},[2714],[507,42490,573],{"className":42491},[2923],[507,42493],{"className":42494,"style":2919},[2714],[507,42496,42498,42501,42504,42507],{"className":42497},[2561],[507,42499],{"className":42500,"style":7035},[2565],[507,42502,584],{"className":42503},[2570],[507,42505,8563],{"className":42506,"style":2776},[2570,2611],[507,42508,42510,42513],{"className":42509},[2570],[507,42511,22278],{"className":42512,"style":27338},[2570,2611],[507,42514,42516],{"className":42515},[2579],[507,42517,42519,42539],{"className":42518},[2583,3200],[507,42520,42522,42536],{"className":42521},[2587],[507,42523,42525],{"className":42524,"style":14281},[2591],[507,42526,42527,42530],{"style":42267},[507,42528],{"className":42529,"style":2600},[2599],[507,42531,42533],{"className":42532},[2604,2605,2606,2607],[507,42534,601],{"className":42535},[2570,2607],[507,42537,3225],{"className":42538},[3224],[507,42540,42542],{"className":42541},[2587],[507,42543,42545],{"className":42544,"style":3232},[2591],[507,42546],{},[41858,42548,42549,42580],{},[41873,42550,42551],{},[507,42552,42554,42568],{"className":42553},[2523],[507,42555,42557],{"className":42556},[2527],[2529,42558,42559],{"xmlns":2531},[2533,42560,42561,42565],{},[2536,42562,42563],{},[2542,42564,42346],{},[2549,42566,42567],{"encoding":2551},"\\omega",[507,42569,42571],{"className":42570,"ariaHidden":2557},[2556],[507,42572,42574,42577],{"className":42573},[2561],[507,42575],{"className":42576,"style":2639},[2565],[507,42578,42346],{"className":42579,"style":2776},[2570,2611],[41873,42581,42582,42583],{},"Angular drive frequency, ",[507,42584,42586,42608],{"className":42585},[2523],[507,42587,42589],{"className":42588},[2527],[2529,42590,42591],{"xmlns":2531},[2533,42592,42593,42605],{},[2536,42594,42595,42597,42599,42601,42603],{},[2542,42596,42346],{},[2689,42598,573],{},[2693,42600,584],{},[2542,42602,8563],{},[2542,42604,22278],{},[2549,42606,42607],{"encoding":2551},"\\omega=2\\pi f",[507,42609,42611,42629],{"className":42610,"ariaHidden":2557},[2556],[507,42612,42614,42617,42620,42623,42626],{"className":42613},[2561],[507,42615],{"className":42616,"style":2639},[2565],[507,42618,42346],{"className":42619,"style":2776},[2570,2611],[507,42621],{"className":42622,"style":2919},[2714],[507,42624,573],{"className":42625},[2923],[507,42627],{"className":42628,"style":2919},[2714],[507,42630,42632,42635,42638,42641],{"className":42631},[2561],[507,42633],{"className":42634,"style":7035},[2565],[507,42636,584],{"className":42637},[2570],[507,42639,8563],{"className":42640,"style":2776},[2570,2611],[507,42642,22278],{"className":42643,"style":27338},[2570,2611],[41858,42645,42646,42678],{},[41873,42647,42648],{},[507,42649,42651,42666],{"className":42650},[2523],[507,42652,42654],{"className":42653},[2527],[2529,42655,42656],{"xmlns":2531},[2533,42657,42658,42663],{},[2536,42659,42660],{},[2542,42661,42662],{"mathvariant":2748},"Ω",[2549,42664,42665],{"encoding":2551},"\\Omega",[507,42667,42669],{"className":42668,"ariaHidden":2557},[2556],[507,42670,42672,42675],{"className":42671},[2561],[507,42673],{"className":42674,"style":2566},[2565],[507,42676,42662],{"className":42677},[2570],[41873,42679,42680],{},"On-resonance Rabi angular frequency",[41858,42682,42683,42724],{},[41873,42684,42685],{},[507,42686,42688,42708],{"className":42687},[2523],[507,42689,42691],{"className":42690},[2527],[2529,42692,42693],{"xmlns":2531},[2533,42694,42695,42705],{},[2536,42696,42697,42699,42701,42703],{},[2542,42698,42662],{"mathvariant":2748},[2542,42700,645],{"mathvariant":2748},[2693,42702,584],{},[2542,42704,8563],{},[2549,42706,42707],{"encoding":2551},"\\Omega\u002F2\\pi",[507,42709,42711],{"className":42710,"ariaHidden":2557},[2556],[507,42712,42714,42717,42721],{"className":42713},[2561],[507,42715],{"className":42716,"style":2769},[2565],[507,42718,42720],{"className":42719},[2570],"Ω\u002F2",[507,42722,8563],{"className":42723,"style":2776},[2570,2611],[41873,42725,42726],{},"On-resonance Rabi frequency in ordinary frequency units",[41858,42728,42729,42761],{},[41873,42730,42731],{},[507,42732,42734,42749],{"className":42733},[2523],[507,42735,42737],{"className":42736},[2527],[2529,42738,42739],{"xmlns":2531},[2533,42740,42741,42746],{},[2536,42742,42743],{},[2542,42744,42745],{"mathvariant":2748},"Δ",[2549,42747,42748],{"encoding":2551},"\\Delta",[507,42750,42752],{"className":42751,"ariaHidden":2557},[2556],[507,42753,42755,42758],{"className":42754},[2561],[507,42756],{"className":42757,"style":2566},[2565],[507,42759,42745],{"className":42760},[2570],[41873,42762,42763,42764],{},"Drive detuning, ",[507,42765,42767,42793],{"className":42766},[2523],[507,42768,42770],{"className":42769},[2527],[2529,42771,42772],{"xmlns":2531},[2533,42773,42774,42790],{},[2536,42775,42776,42778,42780,42782,42784],{},[2542,42777,42745],{"mathvariant":2748},[2689,42779,573],{},[2542,42781,42346],{},[2689,42783,2691],{},[3168,42785,42786,42788],{},[2542,42787,42346],{},[2693,42789,601],{},[2549,42791,42792],{"encoding":2551},"\\Delta=\\omega-\\omega_0",[507,42794,42796,42814,42832],{"className":42795,"ariaHidden":2557},[2556],[507,42797,42799,42802,42805,42808,42811],{"className":42798},[2561],[507,42800],{"className":42801,"style":2566},[2565],[507,42803,42745],{"className":42804},[2570],[507,42806],{"className":42807,"style":2919},[2714],[507,42809,573],{"className":42810},[2923],[507,42812],{"className":42813,"style":2919},[2714],[507,42815,42817,42820,42823,42826,42829],{"className":42816},[2561],[507,42818],{"className":42819,"style":2707},[2565],[507,42821,42346],{"className":42822,"style":2776},[2570,2611],[507,42824],{"className":42825,"style":2715},[2714],[507,42827,2691],{"className":42828},[2719],[507,42830],{"className":42831,"style":2715},[2714],[507,42833,42835,42838],{"className":42834},[2561],[507,42836],{"className":42837,"style":25197},[2565],[507,42839,42841,42844],{"className":42840},[2570],[507,42842,42346],{"className":42843,"style":2776},[2570,2611],[507,42845,42847],{"className":42846},[2579],[507,42848,42850,42870],{"className":42849},[2583,3200],[507,42851,42853,42867],{"className":42852},[2587],[507,42854,42856],{"className":42855,"style":14281},[2591],[507,42857,42858,42861],{"style":4572},[507,42859],{"className":42860,"style":2600},[2599],[507,42862,42864],{"className":42863},[2604,2605,2606,2607],[507,42865,601],{"className":42866},[2570,2607],[507,42868,3225],{"className":42869},[3224],[507,42871,42873],{"className":42872},[2587],[507,42874,42876],{"className":42875,"style":3232},[2591],[507,42877],{},[41858,42879,42880,42952],{},[41873,42881,42882],{},[507,42883,42885,42903],{"className":42884},[2523],[507,42886,42888],{"className":42887},[2527],[2529,42889,42890],{"xmlns":2531},[2533,42891,42892,42900],{},[2536,42893,42894],{},[3168,42895,42896,42898],{},[2542,42897,42662],{"mathvariant":2748},[2542,42899,20370],{},[2549,42901,42902],{"encoding":2551},"\\Omega_R",[507,42904,42906],{"className":42905,"ariaHidden":2557},[2556],[507,42907,42909,42912],{"className":42908},[2561],[507,42910],{"className":42911,"style":3187},[2565],[507,42913,42915,42918],{"className":42914},[2570],[507,42916,42662],{"className":42917},[2570],[507,42919,42921],{"className":42920},[2579],[507,42922,42924,42944],{"className":42923},[2583,3200],[507,42925,42927,42941],{"className":42926},[2587],[507,42928,42930],{"className":42929,"style":3207},[2591],[507,42931,42932,42935],{"style":5398},[507,42933],{"className":42934,"style":2600},[2599],[507,42936,42938],{"className":42937},[2604,2605,2606,2607],[507,42939,20370],{"className":42940,"style":20395},[2570,2611,2607],[507,42942,3225],{"className":42943},[3224],[507,42945,42947],{"className":42946},[2587],[507,42948,42950],{"className":42949,"style":3232},[2591],[507,42951],{},[41873,42953,42954],{},"Generalized Rabi angular frequency",[41858,42956,42957,43032],{},[41873,42958,42959],{},[507,42960,42962,42981],{"className":42961},[2523],[507,42963,42965],{"className":42964},[2527],[2529,42966,42967],{"xmlns":2531},[2533,42968,42969,42978],{},[2536,42970,42971],{},[3168,42972,42973,42976],{},[2542,42974,42975],{},"ν",[2542,42977,20370],{},[2549,42979,42980],{"encoding":2551},"\\nu_R",[507,42982,42984],{"className":42983,"ariaHidden":2557},[2556],[507,42985,42987,42990],{"className":42986},[2561],[507,42988],{"className":42989,"style":25197},[2565],[507,42991,42993,42997],{"className":42992},[2570],[507,42994,42975],{"className":42995,"style":42996},[2570,2611],"margin-right:0.0637em;",[507,42998,43000],{"className":42999},[2579],[507,43001,43003,43024],{"className":43002},[2583,3200],[507,43004,43006,43021],{"className":43005},[2587],[507,43007,43009],{"className":43008,"style":3207},[2591],[507,43010,43012,43015],{"style":43011},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[507,43013],{"className":43014,"style":2600},[2599],[507,43016,43018],{"className":43017},[2604,2605,2606,2607],[507,43019,20370],{"className":43020,"style":20395},[2570,2611,2607],[507,43022,3225],{"className":43023},[3224],[507,43025,43027],{"className":43026},[2587],[507,43028,43030],{"className":43029,"style":3232},[2591],[507,43031],{},[41873,43033,43034,43035],{},"Generalized Rabi frequency, ",[507,43036,43038,43070],{"className":43037},[2523],[507,43039,43041],{"className":43040},[2527],[2529,43042,43043],{"xmlns":2531},[2533,43044,43045,43067],{},[2536,43046,43047,43053,43055,43061,43063,43065],{},[3168,43048,43049,43051],{},[2542,43050,42975],{},[2542,43052,20370],{},[2689,43054,573],{},[3168,43056,43057,43059],{},[2542,43058,42662],{"mathvariant":2748},[2542,43060,20370],{},[2542,43062,645],{"mathvariant":2748},[2693,43064,584],{},[2542,43066,8563],{},[2549,43068,43069],{"encoding":2551},"\\nu_R=\\Omega_R\u002F2\\pi",[507,43071,43073,43128],{"className":43072,"ariaHidden":2557},[2556],[507,43074,43076,43079,43119,43122,43125],{"className":43075},[2561],[507,43077],{"className":43078,"style":25197},[2565],[507,43080,43082,43085],{"className":43081},[2570],[507,43083,42975],{"className":43084,"style":42996},[2570,2611],[507,43086,43088],{"className":43087},[2579],[507,43089,43091,43111],{"className":43090},[2583,3200],[507,43092,43094,43108],{"className":43093},[2587],[507,43095,43097],{"className":43096,"style":3207},[2591],[507,43098,43099,43102],{"style":43011},[507,43100],{"className":43101,"style":2600},[2599],[507,43103,43105],{"className":43104},[2604,2605,2606,2607],[507,43106,20370],{"className":43107,"style":20395},[2570,2611,2607],[507,43109,3225],{"className":43110},[3224],[507,43112,43114],{"className":43113},[2587],[507,43115,43117],{"className":43116,"style":3232},[2591],[507,43118],{},[507,43120],{"className":43121,"style":2919},[2714],[507,43123,573],{"className":43124},[2923],[507,43126],{"className":43127,"style":2919},[2714],[507,43129,43131,43134,43174,43177],{"className":43130},[2561],[507,43132],{"className":43133,"style":2769},[2565],[507,43135,43137,43140],{"className":43136},[2570],[507,43138,42662],{"className":43139},[2570],[507,43141,43143],{"className":43142},[2579],[507,43144,43146,43166],{"className":43145},[2583,3200],[507,43147,43149,43163],{"className":43148},[2587],[507,43150,43152],{"className":43151,"style":3207},[2591],[507,43153,43154,43157],{"style":5398},[507,43155],{"className":43156,"style":2600},[2599],[507,43158,43160],{"className":43159},[2604,2605,2606,2607],[507,43161,20370],{"className":43162,"style":20395},[2570,2611,2607],[507,43164,3225],{"className":43165},[3224],[507,43167,43169],{"className":43168},[2587],[507,43170,43172],{"className":43171,"style":3232},[2591],[507,43173],{},[507,43175,26121],{"className":43176},[2570],[507,43178,8563],{"className":43179,"style":2776},[2570,2611],[41858,43181,43182,43212],{},[41873,43183,43184],{},[507,43185,43187,43200],{"className":43186},[2523],[507,43188,43190],{"className":43189},[2527],[2529,43191,43192],{"xmlns":2531},[2533,43193,43194,43198],{},[2536,43195,43196],{},[2542,43197,41704],{},[2549,43199,41830],{"encoding":2551},[507,43201,43203],{"className":43202,"ariaHidden":2557},[2556],[507,43204,43206,43209],{"className":43205},[2561],[507,43207],{"className":43208,"style":2639},[2565],[507,43210,41704],{"className":43211,"style":41776},[2570,2611],[41873,43213,43214],{},"Pulse duration",[41858,43216,43217,43305],{},[41873,43218,43219],{},[507,43220,43222,43242],{"className":43221},[2523],[507,43223,43225],{"className":43224},[2527],[2529,43226,43227],{"xmlns":2531},[2533,43228,43229,43239],{},[2536,43230,43231],{},[3775,43232,43233,43235,43237],{},[2542,43234,37046],{},[2693,43236,584],{},[2689,43238,20769],{},[2549,43240,43241],{"encoding":2551},"T_2^\\ast",[507,43243,43245],{"className":43244,"ariaHidden":2557},[2556],[507,43246,43248,43252],{"className":43247},[2561],[507,43249],{"className":43250,"style":43251},[2565],"height:0.9368em;vertical-align:-0.2481em;",[507,43253,43255,43258],{"className":43254},[2570],[507,43256,37046],{"className":43257,"style":3220},[2570,2611],[507,43259,43261],{"className":43260},[2579],[507,43262,43264,43296],{"className":43263},[2583,3200],[507,43265,43267,43293],{"className":43266},[2587],[507,43268,43270,43282],{"className":43269,"style":36582},[2591],[507,43271,43273,43276],{"style":43272},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[507,43274],{"className":43275,"style":2600},[2599],[507,43277,43279],{"className":43278},[2604,2605,2606,2607],[507,43280,584],{"className":43281},[2570,2607],[507,43283,43284,43287],{"style":2595},[507,43285],{"className":43286,"style":2600},[2599],[507,43288,43290],{"className":43289},[2604,2605,2606,2607],[507,43291,20769],{"className":43292},[2719,2607],[507,43294,3225],{"className":43295},[3224],[507,43297,43299],{"className":43298},[2587],[507,43300,43303],{"className":43301,"style":43302},[2591],"height:0.2481em;",[507,43304],{},[41873,43306,43307],{},"Phenomenological dephasing time used as an optional damping parameter",[41858,43309,43310,43313],{},[41873,43311,43312],{},"RWA",[41873,43314,43315],{},"Rotating-wave approximation",[41858,43317,43318,43321],{},[41873,43319,43320],{},"FFT",[41873,43322,43323],{},"Fast Fourier transform",[13,43325,43327],{"id":43326},"physical-model","Physical Model",[18,43329,43330,43331,43369,43370,43408],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[507,43332,43334,43351],{"className":43333},[2523],[507,43335,43337],{"className":43336},[2527],[2529,43338,43339],{"xmlns":2531},[2533,43340,43341,43349],{},[2536,43342,43343,43345,43347],{},[2689,43344,2749],{"stretchy":2755},[2542,43346,37913],{},[2689,43348,2756],{"stretchy":2755},[2549,43350,41895],{"encoding":2551},[507,43352,43354],{"className":43353,"ariaHidden":2557},[2556],[507,43355,43357,43360,43363,43366],{"className":43356},[2561],[507,43358],{"className":43359,"style":2769},[2565],[507,43361,2749],{"className":43362},[2941],[507,43364,37913],{"className":43365,"style":2776},[2570,2611],[507,43367,2756],{"className":43368},[2780]," and an excited state ",[507,43371,43373,43390],{"className":43372},[2523],[507,43374,43376],{"className":43375},[2527],[2529,43377,43378],{"xmlns":2531},[2533,43379,43380,43388],{},[2536,43381,43382,43384,43386],{},[2689,43383,2749],{"stretchy":2755},[2542,43385,3286],{},[2689,43387,2756],{"stretchy":2755},[2549,43389,41941],{"encoding":2551},[507,43391,43393],{"className":43392,"ariaHidden":2557},[2556],[507,43394,43396,43399,43402,43405],{"className":43395},[2561],[507,43397],{"className":43398,"style":2769},[2565],[507,43400,2749],{"className":43401},[2941],[507,43403,3286],{"className":43404},[2570,2611],[507,43406,2756],{"className":43407},[2780],". In the absence of a microwave drive, the energy splitting is",[507,43410,43412],{"className":43411},[2784],[507,43413,43415,43454],{"className":43414},[2523],[507,43416,43418],{"className":43417},[2527],[2529,43419,43420],{"xmlns":2531,"display":2793},[2533,43421,43422,43451],{},[2536,43423,43424,43430,43432,43438,43440,43443,43449],{},[3168,43425,43426,43428],{},[2542,43427,6182],{},[2542,43429,3286],{},[2689,43431,2691],{},[3168,43433,43434,43436],{},[2542,43435,6182],{},[2542,43437,37913],{},[2689,43439,573],{},[2542,43441,43442],{"mathvariant":2748},"ℏ",[3168,43444,43445,43447],{},[2542,43446,42346],{},[2693,43448,601],{},[2689,43450,2819],{"separator":2557},[2549,43452,43453],{"encoding":2551},"E_e-E_g=\\hbar\\omega_0,",[507,43455,43457,43512,43567],{"className":43456,"ariaHidden":2557},[2556],[507,43458,43460,43463,43503,43506,43509],{"className":43459},[2561],[507,43461],{"className":43462,"style":3187},[2565],[507,43464,43466,43469],{"className":43465},[2570],[507,43467,6182],{"className":43468,"style":5892},[2570,2611],[507,43470,43472],{"className":43471},[2579],[507,43473,43475,43495],{"className":43474},[2583,3200],[507,43476,43478,43492],{"className":43477},[2587],[507,43479,43481],{"className":43480,"style":4507},[2591],[507,43482,43483,43486],{"style":6014},[507,43484],{"className":43485,"style":2600},[2599],[507,43487,43489],{"className":43488},[2604,2605,2606,2607],[507,43490,3286],{"className":43491},[2570,2611,2607],[507,43493,3225],{"className":43494},[3224],[507,43496,43498],{"className":43497},[2587],[507,43499,43501],{"className":43500,"style":3232},[2591],[507,43502],{},[507,43504],{"className":43505,"style":2715},[2714],[507,43507,2691],{"className":43508},[2719],[507,43510],{"className":43511,"style":2715},[2714],[507,43513,43515,43518,43558,43561,43564],{"className":43514},[2561],[507,43516],{"className":43517,"style":7967},[2565],[507,43519,43521,43524],{"className":43520},[2570],[507,43522,6182],{"className":43523,"style":5892},[2570,2611],[507,43525,43527],{"className":43526},[2579],[507,43528,43530,43550],{"className":43529},[2583,3200],[507,43531,43533,43547],{"className":43532},[2587],[507,43534,43536],{"className":43535,"style":4507},[2591],[507,43537,43538,43541],{"style":6014},[507,43539],{"className":43540,"style":2600},[2599],[507,43542,43544],{"className":43543},[2604,2605,2606,2607],[507,43545,37913],{"className":43546,"style":2776},[2570,2611,2607],[507,43548,3225],{"className":43549},[3224],[507,43551,43553],{"className":43552},[2587],[507,43554,43556],{"className":43555,"style":6833},[2591],[507,43557],{},[507,43559],{"className":43560,"style":2919},[2714],[507,43562,573],{"className":43563},[2923],[507,43565],{"className":43566,"style":2919},[2714],[507,43568,43570,43574,43577,43617],{"className":43569},[2561],[507,43571],{"className":43572,"style":43573},[2565],"height:0.8833em;vertical-align:-0.1944em;",[507,43575,43442],{"className":43576},[2570],[507,43578,43580,43583],{"className":43579},[2570],[507,43581,42346],{"className":43582,"style":2776},[2570,2611],[507,43584,43586],{"className":43585},[2579],[507,43587,43589,43609],{"className":43588},[2583,3200],[507,43590,43592,43606],{"className":43591},[2587],[507,43593,43595],{"className":43594,"style":14281},[2591],[507,43596,43597,43600],{"style":4572},[507,43598],{"className":43599,"style":2600},[2599],[507,43601,43603],{"className":43602},[2604,2605,2606,2607],[507,43604,601],{"className":43605},[2570,2607],[507,43607,3225],{"className":43608},[3224],[507,43610,43612],{"className":43611},[2587],[507,43613,43615],{"className":43614,"style":3232},[2591],[507,43616],{},[507,43618,2819],{"className":43619},[2961],[18,43621,43622],{},"where the angular transition frequency is",[507,43624,43626],{"className":43625},[2784],[507,43627,43629,43661],{"className":43628},[2523],[507,43630,43632],{"className":43631},[2527],[2529,43633,43634],{"xmlns":2531,"display":2793},[2533,43635,43636,43658],{},[2536,43637,43638,43644,43646,43648,43650,43656],{},[3168,43639,43640,43642],{},[2542,43641,42346],{},[2693,43643,601],{},[2689,43645,573],{},[2693,43647,584],{},[2542,43649,8563],{},[3168,43651,43652,43654],{},[2542,43653,22278],{},[2693,43655,601],{},[2542,43657,53],{"mathvariant":2748},[2549,43659,43660],{"encoding":2551},"\\omega_0=2\\pi f_0.",[507,43662,43664,43719],{"className":43663,"ariaHidden":2557},[2556],[507,43665,43667,43670,43710,43713,43716],{"className":43666},[2561],[507,43668],{"className":43669,"style":25197},[2565],[507,43671,43673,43676],{"className":43672},[2570],[507,43674,42346],{"className":43675,"style":2776},[2570,2611],[507,43677,43679],{"className":43678},[2579],[507,43680,43682,43702],{"className":43681},[2583,3200],[507,43683,43685,43699],{"className":43684},[2587],[507,43686,43688],{"className":43687,"style":14281},[2591],[507,43689,43690,43693],{"style":4572},[507,43691],{"className":43692,"style":2600},[2599],[507,43694,43696],{"className":43695},[2604,2605,2606,2607],[507,43697,601],{"className":43698},[2570,2607],[507,43700,3225],{"className":43701},[3224],[507,43703,43705],{"className":43704},[2587],[507,43706,43708],{"className":43707,"style":3232},[2591],[507,43709],{},[507,43711],{"className":43712,"style":2919},[2714],[507,43714,573],{"className":43715},[2923],[507,43717],{"className":43718,"style":2919},[2714],[507,43720,43722,43725,43728,43731,43771],{"className":43721},[2561],[507,43723],{"className":43724,"style":7035},[2565],[507,43726,584],{"className":43727},[2570],[507,43729,8563],{"className":43730,"style":2776},[2570,2611],[507,43732,43734,43737],{"className":43733},[2570],[507,43735,22278],{"className":43736,"style":27338},[2570,2611],[507,43738,43740],{"className":43739},[2579],[507,43741,43743,43763],{"className":43742},[2583,3200],[507,43744,43746,43760],{"className":43745},[2587],[507,43747,43749],{"className":43748,"style":14281},[2591],[507,43750,43751,43754],{"style":42267},[507,43752],{"className":43753,"style":2600},[2599],[507,43755,43757],{"className":43756},[2604,2605,2606,2607],[507,43758,601],{"className":43759},[2570,2607],[507,43761,3225],{"className":43762},[3224],[507,43764,43766],{"className":43765},[2587],[507,43767,43769],{"className":43768,"style":3232},[2591],[507,43770],{},[507,43772,53],{"className":43773},[2570],[18,43775,43776],{},"The default configuration uses a transition frequency of",[507,43778,43780],{"className":43779},[2784],[507,43781,43783,43819],{"className":43782},[2523],[507,43784,43786],{"className":43785},[2527],[2529,43787,43788],{"xmlns":2531,"display":2793},[2533,43789,43790,43816],{},[2536,43791,43792,43798,43800,43803,43805,43814],{},[3168,43793,43794,43796],{},[2542,43795,22278],{},[2693,43797,601],{},[2689,43799,573],{},[2693,43801,43802],{},"5.000",[6167,43804,6169],{},[2536,43806,43807,43810,43812],{},[2542,43808,43809],{"mathvariant":2748},"G",[2542,43811,3138],{"mathvariant":2748},[2542,43813,666],{"mathvariant":2748},[2542,43815,53],{"mathvariant":2748},[2549,43817,43818],{"encoding":2551},"f_0=5.000~\\mathrm{GHz}.",[507,43820,43822,43877],{"className":43821,"ariaHidden":2557},[2556],[507,43823,43825,43828,43868,43871,43874],{"className":43824},[2561],[507,43826],{"className":43827,"style":7035},[2565],[507,43829,43831,43834],{"className":43830},[2570],[507,43832,22278],{"className":43833,"style":27338},[2570,2611],[507,43835,43837],{"className":43836},[2579],[507,43838,43840,43860],{"className":43839},[2583,3200],[507,43841,43843,43857],{"className":43842},[2587],[507,43844,43846],{"className":43845,"style":14281},[2591],[507,43847,43848,43851],{"style":42267},[507,43849],{"className":43850,"style":2600},[2599],[507,43852,43854],{"className":43853},[2604,2605,2606,2607],[507,43855,601],{"className":43856},[2570,2607],[507,43858,3225],{"className":43859},[3224],[507,43861,43863],{"className":43862},[2587],[507,43864,43866],{"className":43865,"style":3232},[2591],[507,43867],{},[507,43869],{"className":43870,"style":2919},[2714],[507,43872,573],{"className":43873},[2923],[507,43875],{"className":43876,"style":2919},[2714],[507,43878,43880,43883,43886,43890,43898],{"className":43879},[2561],[507,43881],{"className":43882,"style":2566},[2565],[507,43884,43802],{"className":43885},[2570],[507,43887,6169],{"className":43888},[2714,43889],"nobreak",[507,43891,43893],{"className":43892},[2570],[507,43894,43897],{"className":43895},[2570,43896],"mathrm","GHz",[507,43899,53],{"className":43900},[2570],[18,43902,43903,43904,43932,43933,10799,43971,44009],{},"Applying a microwave drive with frequency ",[507,43905,43907,43920],{"className":43906},[2523],[507,43908,43910],{"className":43909},[2527],[2529,43911,43912],{"xmlns":2531},[2533,43913,43914,43918],{},[2536,43915,43916],{},[2542,43917,22278],{},[2549,43919,22278],{"encoding":2551},[507,43921,43923],{"className":43922,"ariaHidden":2557},[2556],[507,43924,43926,43929],{"className":43925},[2561],[507,43927],{"className":43928,"style":7035},[2565],[507,43930,22278],{"className":43931,"style":27338},[2570,2611]," induces coherent rotations between ",[507,43934,43936,43953],{"className":43935},[2523],[507,43937,43939],{"className":43938},[2527],[2529,43940,43941],{"xmlns":2531},[2533,43942,43943,43951],{},[2536,43944,43945,43947,43949],{},[2689,43946,2749],{"stretchy":2755},[2542,43948,37913],{},[2689,43950,2756],{"stretchy":2755},[2549,43952,41895],{"encoding":2551},[507,43954,43956],{"className":43955,"ariaHidden":2557},[2556],[507,43957,43959,43962,43965,43968],{"className":43958},[2561],[507,43960],{"className":43961,"style":2769},[2565],[507,43963,2749],{"className":43964},[2941],[507,43966,37913],{"className":43967,"style":2776},[2570,2611],[507,43969,2756],{"className":43970},[2780],[507,43972,43974,43991],{"className":43973},[2523],[507,43975,43977],{"className":43976},[2527],[2529,43978,43979],{"xmlns":2531},[2533,43980,43981,43989],{},[2536,43982,43983,43985,43987],{},[2689,43984,2749],{"stretchy":2755},[2542,43986,3286],{},[2689,43988,2756],{"stretchy":2755},[2549,43990,41941],{"encoding":2551},[507,43992,43994],{"className":43993,"ariaHidden":2557},[2556],[507,43995,43997,44000,44003,44006],{"className":43996},[2561],[507,43998],{"className":43999,"style":2769},[2565],[507,44001,2749],{"className":44002},[2941],[507,44004,3286],{"className":44005},[2570,2611],[507,44007,2756],{"className":44008},[2780],". Exact resonance occurs when",[507,44011,44013],{"className":44012},[2784],[507,44014,44016,44040],{"className":44015},[2523],[507,44017,44019],{"className":44018},[2527],[2529,44020,44021],{"xmlns":2531,"display":2793},[2533,44022,44023,44037],{},[2536,44024,44025,44027,44029,44035],{},[2542,44026,22278],{},[2689,44028,573],{},[3168,44030,44031,44033],{},[2542,44032,22278],{},[2693,44034,601],{},[2542,44036,53],{"mathvariant":2748},[2549,44038,44039],{"encoding":2551},"f=f_0.",[507,44041,44043,44061],{"className":44042,"ariaHidden":2557},[2556],[507,44044,44046,44049,44052,44055,44058],{"className":44045},[2561],[507,44047],{"className":44048,"style":7035},[2565],[507,44050,22278],{"className":44051,"style":27338},[2570,2611],[507,44053],{"className":44054,"style":2919},[2714],[507,44056,573],{"className":44057},[2923],[507,44059],{"className":44060,"style":2919},[2714],[507,44062,44064,44067,44107],{"className":44063},[2561],[507,44065],{"className":44066,"style":7035},[2565],[507,44068,44070,44073],{"className":44069},[2570],[507,44071,22278],{"className":44072,"style":27338},[2570,2611],[507,44074,44076],{"className":44075},[2579],[507,44077,44079,44099],{"className":44078},[2583,3200],[507,44080,44082,44096],{"className":44081},[2587],[507,44083,44085],{"className":44084,"style":14281},[2591],[507,44086,44087,44090],{"style":42267},[507,44088],{"className":44089,"style":2600},[2599],[507,44091,44093],{"className":44092},[2604,2605,2606,2607],[507,44094,601],{"className":44095},[2570,2607],[507,44097,3225],{"className":44098},[3224],[507,44100,44102],{"className":44101},[2587],[507,44103,44105],{"className":44104,"style":3232},[2591],[507,44106],{},[507,44108,53],{"className":44109},[2570],[18,44111,44112],{},"At resonance, the drive produces ideal Rabi oscillations. Away from resonance, detuning changes the oscillation frequency and reduces the maximum achievable excited-state probability. This same two-level control structure appears across superconducting qubits, spin qubits, trapped ions, neutral atoms, and nitrogen-vacancy centers whenever a selected transition can be treated as an effective qubit.",[13,44114,44116],{"id":44115},"mathematical-foundation","Mathematical Foundation",[18,44118,44119],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[507,44121,44123],{"className":44122},[2784],[507,44124,44126,44166],{"className":44125},[2523],[507,44127,44129],{"className":44128},[2527],[2529,44130,44131],{"xmlns":2531,"display":2793},[2533,44132,44133,44163],{},[2536,44134,44135,44137,44139,44141,44143,44145,44147,44149,44151,44153,44155,44157,44159,44161],{},[2689,44136,2749],{"stretchy":2755},[2542,44138,3793],{},[2689,44140,2756],{"stretchy":2755},[2689,44142,573],{},[2542,44144,42027],{},[2689,44146,2749],{"stretchy":2755},[2542,44148,37913],{},[2689,44150,2756],{"stretchy":2755},[2689,44152,2107],{},[2542,44154,36665],{},[2689,44156,2749],{"stretchy":2755},[2542,44158,3286],{},[2689,44160,2756],{"stretchy":2755},[2689,44162,2819],{"separator":2557},[2549,44164,44165],{"encoding":2551},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[507,44167,44169,44193,44220],{"className":44168,"ariaHidden":2557},[2556],[507,44170,44172,44175,44178,44181,44184,44187,44190],{"className":44171},[2561],[507,44173],{"className":44174,"style":2769},[2565],[507,44176,2749],{"className":44177},[2941],[507,44179,3793],{"className":44180,"style":2776},[2570,2611],[507,44182,2756],{"className":44183},[2780],[507,44185],{"className":44186,"style":2919},[2714],[507,44188,573],{"className":44189},[2923],[507,44191],{"className":44192,"style":2919},[2714],[507,44194,44196,44199,44202,44205,44208,44211,44214,44217],{"className":44195},[2561],[507,44197],{"className":44198,"style":2769},[2565],[507,44200,42027],{"className":44201,"style":42047},[2570,2611],[507,44203,2749],{"className":44204},[2941],[507,44206,37913],{"className":44207,"style":2776},[2570,2611],[507,44209,2756],{"className":44210},[2780],[507,44212],{"className":44213,"style":2715},[2714],[507,44215,2107],{"className":44216},[2719],[507,44218],{"className":44219,"style":2715},[2714],[507,44221,44223,44226,44229,44232,44235,44238],{"className":44222},[2561],[507,44224],{"className":44225,"style":2769},[2565],[507,44227,36665],{"className":44228,"style":36760},[2570,2611],[507,44230,2749],{"className":44231},[2941],[507,44233,3286],{"className":44234},[2570,2611],[507,44236,2756],{"className":44237},[2780],[507,44239,2819],{"className":44240},[2961],[18,44242,28454,44243,10799,44272,44301],{},[507,44244,44246,44260],{"className":44245},[2523],[507,44247,44249],{"className":44248},[2527],[2529,44250,44251],{"xmlns":2531},[2533,44252,44253,44257],{},[2536,44254,44255],{},[2542,44256,42027],{},[2549,44258,44259],{"encoding":2551},"\\alpha",[507,44261,44263],{"className":44262,"ariaHidden":2557},[2556],[507,44264,44266,44269],{"className":44265},[2561],[507,44267],{"className":44268,"style":2639},[2565],[507,44270,42027],{"className":44271,"style":42047},[2570,2611],[507,44273,44275,44289],{"className":44274},[2523],[507,44276,44278],{"className":44277},[2527],[2529,44279,44280],{"xmlns":2531},[2533,44281,44282,44286],{},[2536,44283,44284],{},[2542,44285,36665],{},[2549,44287,44288],{"encoding":2551},"\\beta",[507,44290,44292],{"className":44291,"ariaHidden":2557},[2556],[507,44293,44295,44298],{"className":44294},[2561],[507,44296],{"className":44297,"style":7035},[2565],[507,44299,36665],{"className":44300,"style":36760},[2570,2611]," are complex probability amplitudes. Normalization requires",[507,44303,44305],{"className":44304},[2784],[507,44306,44308,44347],{"className":44307},[2523],[507,44309,44311],{"className":44310},[2527],[2529,44312,44313],{"xmlns":2531,"display":2793},[2533,44314,44315,44344],{},[2536,44316,44317,44319,44321,44327,44329,44331,44333,44339,44341],{},[2542,44318,2749],{"mathvariant":2748},[2542,44320,42027],{},[2539,44322,44323,44325],{},[2542,44324,2749],{"mathvariant":2748},[2693,44326,584],{},[2689,44328,2107],{},[2542,44330,2749],{"mathvariant":2748},[2542,44332,36665],{},[2539,44334,44335,44337],{},[2542,44336,2749],{"mathvariant":2748},[2693,44338,584],{},[2689,44340,573],{},[2693,44342,44343],{},"1.",[2549,44345,44346],{"encoding":2551},"|\\alpha|^2+|\\beta|^2=1.",[507,44348,44350,44401,44451],{"className":44349,"ariaHidden":2557},[2556],[507,44351,44353,44357,44360,44363,44392,44395,44398],{"className":44352},[2561],[507,44354],{"className":44355,"style":44356},[2565],"height:1.1141em;vertical-align:-0.25em;",[507,44358,2749],{"className":44359},[2570],[507,44361,42027],{"className":44362,"style":42047},[2570,2611],[507,44364,44366,44369],{"className":44365},[2570],[507,44367,2749],{"className":44368},[2570],[507,44370,44372],{"className":44371},[2579],[507,44373,44375],{"className":44374},[2583],[507,44376,44378],{"className":44377},[2587],[507,44379,44381],{"className":44380,"style":3002},[2591],[507,44382,44383,44386],{"style":2906},[507,44384],{"className":44385,"style":2600},[2599],[507,44387,44389],{"className":44388},[2604,2605,2606,2607],[507,44390,584],{"className":44391},[2570,2607],[507,44393],{"className":44394,"style":2715},[2714],[507,44396,2107],{"className":44397},[2719],[507,44399],{"className":44400,"style":2715},[2714],[507,44402,44404,44407,44410,44413,44442,44445,44448],{"className":44403},[2561],[507,44405],{"className":44406,"style":44356},[2565],[507,44408,2749],{"className":44409},[2570],[507,44411,36665],{"className":44412,"style":36760},[2570,2611],[507,44414,44416,44419],{"className":44415},[2570],[507,44417,2749],{"className":44418},[2570],[507,44420,44422],{"className":44421},[2579],[507,44423,44425],{"className":44424},[2583],[507,44426,44428],{"className":44427},[2587],[507,44429,44431],{"className":44430,"style":3002},[2591],[507,44432,44433,44436],{"style":2906},[507,44434],{"className":44435,"style":2600},[2599],[507,44437,44439],{"className":44438},[2604,2605,2606,2607],[507,44440,584],{"className":44441},[2570,2607],[507,44443],{"className":44444,"style":2919},[2714],[507,44446,573],{"className":44447},[2923],[507,44449],{"className":44450,"style":2919},[2714],[507,44452,44454,44457],{"className":44453},[2561],[507,44455],{"className":44456,"style":2729},[2565],[507,44458,44343],{"className":44459},[2570],[18,44461,44462],{},"The Born rule gives the probability of measuring the excited state,",[507,44464,44466],{"className":44465},[2784],[507,44467,44469,44521],{"className":44468},[2523],[507,44470,44472],{"className":44471},[2527],[2529,44473,44474],{"xmlns":2531,"display":2793},[2533,44475,44476,44518],{},[2536,44477,44478,44484,44486,44488,44490,44492,44494,44496,44498,44504,44506,44508,44510,44516],{},[3168,44479,44480,44482],{},[2542,44481,3174],{},[2542,44483,3286],{},[2689,44485,573],{},[2542,44487,2749],{"mathvariant":2748},[2689,44489,4425],{"stretchy":2755},[2542,44491,3286],{},[2689,44493,2749],{"stretchy":2755},[2542,44495,3793],{},[2689,44497,2756],{"stretchy":2755},[2539,44499,44500,44502],{},[2542,44501,2749],{"mathvariant":2748},[2693,44503,584],{},[2689,44505,573],{},[2542,44507,2749],{"mathvariant":2748},[2542,44509,36665],{},[2539,44511,44512,44514],{},[2542,44513,2749],{"mathvariant":2748},[2693,44515,584],{},[2542,44517,53],{"mathvariant":2748},[2549,44519,44520],{"encoding":2551},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[507,44522,44524,44579,44641],{"className":44523,"ariaHidden":2557},[2556],[507,44525,44527,44530,44570,44573,44576],{"className":44526},[2561],[507,44528],{"className":44529,"style":3187},[2565],[507,44531,44533,44536],{"className":44532},[2570],[507,44534,3174],{"className":44535,"style":3220},[2570,2611],[507,44537,44539],{"className":44538},[2579],[507,44540,44542,44562],{"className":44541},[2583,3200],[507,44543,44545,44559],{"className":44544},[2587],[507,44546,44548],{"className":44547,"style":4507},[2591],[507,44549,44550,44553],{"style":7637},[507,44551],{"className":44552,"style":2600},[2599],[507,44554,44556],{"className":44555},[2604,2605,2606,2607],[507,44557,3286],{"className":44558},[2570,2611,2607],[507,44560,3225],{"className":44561},[3224],[507,44563,44565],{"className":44564},[2587],[507,44566,44568],{"className":44567,"style":3232},[2591],[507,44569],{},[507,44571],{"className":44572,"style":2919},[2714],[507,44574,573],{"className":44575},[2923],[507,44577],{"className":44578,"style":2919},[2714],[507,44580,44582,44585,44588,44591,44594,44597,44600,44603,44632,44635,44638],{"className":44581},[2561],[507,44583],{"className":44584,"style":44356},[2565],[507,44586,2749],{"className":44587},[2570],[507,44589,4425],{"className":44590},[2941],[507,44592,3286],{"className":44593},[2570,2611],[507,44595,2749],{"className":44596},[2941],[507,44598,3793],{"className":44599,"style":2776},[2570,2611],[507,44601,2756],{"className":44602},[2780],[507,44604,44606,44609],{"className":44605},[2570],[507,44607,2749],{"className":44608},[2570],[507,44610,44612],{"className":44611},[2579],[507,44613,44615],{"className":44614},[2583],[507,44616,44618],{"className":44617},[2587],[507,44619,44621],{"className":44620,"style":3002},[2591],[507,44622,44623,44626],{"style":2906},[507,44624],{"className":44625,"style":2600},[2599],[507,44627,44629],{"className":44628},[2604,2605,2606,2607],[507,44630,584],{"className":44631},[2570,2607],[507,44633],{"className":44634,"style":2919},[2714],[507,44636,573],{"className":44637},[2923],[507,44639],{"className":44640,"style":2919},[2714],[507,44642,44644,44647,44650,44653,44682],{"className":44643},[2561],[507,44645],{"className":44646,"style":44356},[2565],[507,44648,2749],{"className":44649},[2570],[507,44651,36665],{"className":44652,"style":36760},[2570,2611],[507,44654,44656,44659],{"className":44655},[2570],[507,44657,2749],{"className":44658},[2570],[507,44660,44662],{"className":44661},[2579],[507,44663,44665],{"className":44664},[2583],[507,44666,44668],{"className":44667},[2587],[507,44669,44671],{"className":44670,"style":3002},[2591],[507,44672,44673,44676],{"style":2906},[507,44674],{"className":44675,"style":2600},[2599],[507,44677,44679],{"className":44678},[2604,2605,2606,2607],[507,44680,584],{"className":44681},[2570,2607],[507,44683,53],{"className":44684},[2570],[18,44686,44687],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[507,44689,44691],{"className":44690},[2784],[507,44692,44694,44858],{"className":44693},[2523],[507,44695,44697],{"className":44696},[2527],[2529,44698,44699],{"xmlns":2531,"display":2793},[2533,44700,44701,44855],{},[2536,44702,44703,44710,44712,44749,44751,44753,44759,44761,44801,44803,44805,44811,44813,44853],{},[3168,44704,44705,44708],{},[2542,44706,44707],{},"σ",[2542,44709,9139],{},[2689,44711,573],{},[2536,44713,44714,44716,44747],{},[2689,44715,12248],{"fence":2557},[9452,44717,44719,44733],{"rowspacing":9454,"columnalign":44718,"columnspacing":9455},"center 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-1\\end{bmatrix}.",[507,44859,44861,44916,45106,45293],{"className":44860,"ariaHidden":2557},[2556],[507,44862,44864,44867,44907,44910,44913],{"className":44863},[2561],[507,44865],{"className":44866,"style":25197},[2565],[507,44868,44870,44873],{"className":44869},[2570],[507,44871,44707],{"className":44872,"style":2776},[2570,2611],[507,44874,44876],{"className":44875},[2579],[507,44877,44879,44899],{"className":44878},[2583,3200],[507,44880,44882,44896],{"className":44881},[2587],[507,44883,44885],{"className":44884,"style":4507},[2591],[507,44886,44887,44890],{"style":4572},[507,44888],{"className":44889,"style":2600},[2599],[507,44891,44893],{"className":44892},[2604,2605,2606,2607],[507,44894,9139],{"className":44895},[2570,2611,2607],[507,44897,3225],{"className":44898},[3224],[507,44900,44902],{"className":44901},[2587],[507,44903,44905],{"className":44904,"style":3232},[2591],[507,44906],{},[507,44908],{"className":44909,"style":2919},[2714],[507,44911,573],{"className":44912},[2923],[507,44914],{"className":44915,"style":2919},[2714],[507,44917,44919,44922,45045,45048,45051,45054,45057,45097,45100,45103],{"className":44918},[2561],[507,44920],{"className":44921,"style":37668},[2565],[507,44923,44925,44931,45039],{"className":44924},[2937],[507,44926,44928],{"className":44927,"style":2943},[2941,2942],[507,44929,12248],{"className":44930},[2947,2606],[507,44932,44934],{"className":44933},[2570],[507,44935,44937,44986,44991,44994],{"className":44936},[9452],[507,44938,44940],{"className":44939},[9711],[507,44941,44943,44977],{"className":44942},[2583,3200],[507,44944,44946,44974],{"className":44945},[2587],[507,44947,44950,44962],{"className":44948,"style":44949},[2591],"height:1.45em;",[507,44951,44953,44956],{"style":44952},"top:-3.61em;",[507,44954],{"className":44955,"style":4310},[2599],[507,44957,44959],{"className":44958},[2570],[507,44960,601],{"className":44961},[2570],[507,44963,44965,44968],{"style":44964},"top:-2.41em;",[507,44966],{"className":44967,"style":4310},[2599],[507,44969,44971],{"className":44970},[2570],[507,44972,625],{"className":44973},[2570],[507,44975,3225],{"className":44976},[3224],[507,44978,44980],{"className":44979},[2587],[507,44981,44984],{"className":44982,"style":44983},[2591],"height:0.95em;",[507,44985],{},[507,44987],{"className":44988,"style":44990},[44989],"arraycolsep","width:0.5em;",[507,44992],{"className":44993,"style":44990},[44989],[507,44995,44997],{"className":44996},[9711],[507,44998,45000,45031],{"className":44999},[2583,3200],[507,45001,45003,45028],{"className":45002},[2587],[507,45004,45006,45017],{"className":45005,"style":44949},[2591],[507,45007,45008,45011],{"style":44952},[507,45009],{"className":45010,"style":4310},[2599],[507,45012,45014],{"className":45013},[2570],[507,45015,625],{"className":45016},[2570],[507,45018,45019,45022],{"style":44964},[507,45020],{"className":45021,"style":4310},[2599],[507,45023,45025],{"className":45024},[2570],[507,45026,601],{"className":45027},[2570],[507,45029,3225],{"className":45030},[3224],[507,45032,45034],{"className":45033},[2587],[507,45035,45037],{"className":45036,"style":44983},[2591],[507,45038],{},[507,45040,45042],{"className":45041,"style":2943},[2780,2942],[507,45043,12273],{"className":45044},[2947,2606],[507,45046],{"className":45047,"style":2965},[2714],[507,45049,2819],{"className":45050},[2961],[507,45052],{"className":45053,"style":18629},[2714],[507,45055],{"className":45056,"style":2965},[2714],[507,45058,45060,45063],{"className":45059},[2570],[507,45061,44707],{"className":45062,"style":2776},[2570,2611],[507,45064,45066],{"className":45065},[2579],[507,45067,45069,45089],{"className":45068},[2583,3200],[507,45070,45072,45086],{"className":45071},[2587],[507,45073,45075],{"className":45074,"style":4507},[2591],[507,45076,45077,45080],{"style":4572},[507,45078],{"className":45079,"style":2600},[2599],[507,45081,45083],{"className":45082},[2604,2605,2606,2607],[507,45084,20455],{"className":45085,"style":2776},[2570,2611,2607],[507,45087,3225],{"className":45088},[3224],[507,45090,45092],{"className":45091},[2587],[507,45093,45095],{"className":45094,"style":6833},[2591],[507,45096],{},[507,45098],{"className":45099,"style":2919},[2714],[507,45101,573],{"className":45102},[2923],[507,45104],{"className":45105,"style":2919},[2714],[507,45107,45109,45112,45232,45235,45238,45241,45244,45284,45287,45290],{"className":45108},[2561],[507,45110],{"className":45111,"style":37668},[2565],[507,45113,45115,45121,45226],{"className":45114},[2937],[507,45116,45118],{"className":45117,"style":2943},[2941,2942],[507,45119,12248],{"className":45120},[2947,2606],[507,45122,45124],{"className":45123},[2570],[507,45125,45127,45172,45175,45178],{"className":45126},[9452],[507,45128,45130],{"className":45129},[9711],[507,45131,45133,45164],{"className":45132},[2583,3200],[507,45134,45136,45161],{"className":45135},[2587],[507,45137,45139,45150],{"className":45138,"style":44949},[2591],[507,45140,45141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Applied detuning supplies the longitudinal control component.",[13,45463,45465],{"id":45464},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[18,45467,45468],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[507,45470,45472],{"className":45471},[2784],[507,45473,45475,45543],{"className":45474},[2523],[507,45476,45478],{"className":45477},[2527],[2529,45479,45480],{"xmlns":2531,"display":2793},[2533,45481,45482,45540],{},[2536,45483,45484,45486,45488,45490,45492,45494,45508,45514,45516,45518,45520,45522,45524,45526,45528,45530,45532,45538],{},[2542,45485,3138],{},[2689,45487,580],{"stretchy":2755},[2542,45489,3298],{},[2689,45491,3649],{"stretchy":2755},[2689,45493,573],{},[9491,45495,45496,45506],{},[2536,45497,45498,45500],{},[2542,45499,43442],{"mathvariant":2748},[3168,45501,45502,45504],{},[2542,45503,42346],{},[2693,45505,601],{},[2693,45507,584],{},[3168,45509,45510,45512],{},[2542,45511,44707],{},[2542,45513,666],{},[2689,45515,2107],{},[2542,45517,43442],{"mathvariant":2748},[2542,45519,42662],{"mathvariant":2748},[2542,45521,37932],{},[2689,45523,36690],{},[2689,45525,580],{"stretchy":2755},[2542,45527,42346],{},[2542,45529,3298],{},[2689,45531,3649],{"stretchy":2755},[3168,45533,45534,45536],{},[2542,45535,44707],{},[2542,45537,9139],{},[2689,45539,2819],{"separator":2557},[2549,45541,45542],{"encoding":2551},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[507,45544,45546,45573,45732],{"className":45545,"ariaHidden":2557},[2556],[507,45547,45549,45552,45555,45558,45561,45564,45567,45570],{"className":45548},[2561],[507,45550],{"className":45551,"style":2769},[2565],[507,45553,3138],{"className":45554,"style":3153},[2570,2611],[507,45556,580],{"className":45557},[2941],[507,45559,3298],{"className":45560},[2570,2611],[507,45562,3649],{"className":45563},[2780],[507,45565],{"className":45566,"style":2919},[2714],[507,45568,573],{"className":45569},[2923],[507,45571],{"className":45572,"style":2919},[2714],[507,45574,45576,45580,45683,45723,45726,45729],{"className":45575},[2561],[507,45577],{"className":45578,"style":45579},[2565],"height:2.0519em;vertical-align:-0.686em;",[507,45581,45583,45586,45680],{"className":45582},[2570],[507,45584],{"className":45585},[2941,9793],[507,45587,45589],{"className":45588},[9491],[507,45590,45592,45672],{"className":45591},[2583,3200],[507,45593,45595,45669],{"className":45594},[2587],[507,45596,45599,45610,45618],{"className":45597,"style":45598},[2591],"height:1.3659em;",[507,45600,45601,45604],{"style":11717},[507,45602],{"className":45603,"style":4310},[2599],[507,45605,45607],{"className":45606},[2570],[507,45608,584],{"className":45609},[2570],[507,45611,45612,45615],{"style":9878},[507,45613],{"className":45614,"style":4310},[2599],[507,45616],{"className":45617,"style":9886},[9885],[507,45619,45620,45623],{"style":9889},[507,45621],{"className":45622,"style":4310},[2599],[507,45624,45626,45629],{"className":45625},[2570],[507,45627,43442],{"className":45628},[2570],[507,45630,45632,45635],{"className":45631},[2570],[507,45633,42346],{"className":45634,"style":2776},[2570,2611],[507,45636,45638],{"className":45637},[2579],[507,45639,45641,45661],{"className":45640},[2583,3200],[507,45642,45644,45658],{"className":45643},[2587],[507,45645,45647],{"className":45646,"style":14281},[2591],[507,45648,45649,45652],{"style":4572},[507,45650],{"className":45651,"style":2600},[2599],[507,45653,45655],{"className":45654},[2604,2605,2606,2607],[507,45656,601],{"className":45657},[2570,2607],[507,45659,3225],{"className":45660},[3224],[507,45662,45664],{"className":45663},[2587],[507,45665,45667],{"className":45666,"style":3232},[2591],[507,45668],{},[507,45670,3225],{"className":45671},[3224],[507,45673,45675],{"className":45674},[2587],[507,45676,45678],{"className":45677,"style":11755},[2591],[507,45679],{},[507,45681],{"className":45682},[2780,9793],[507,45684,45686,45689],{"className":45685},[2570],[507,45687,44707],{"className":45688,"style":2776},[2570,2611],[507,45690,45692],{"className":45691},[2579],[507,45693,45695,45715],{"className":45694},[2583,3200],[507,45696,45698,45712],{"className":45697},[2587],[507,45699,45701],{"className":45700,"style":4507},[2591],[507,45702,45703,45706],{"style":4572},[507,45704],{"className":45705,"style":2600},[2599],[507,45707,45709],{"className":45708},[2604,2605,2606,2607],[507,45710,666],{"className":45711,"style":20582},[2570,2611,2607],[507,45713,3225],{"className":45714},[3224],[507,45716,45718],{"className":45717},[2587],[507,45719,45721],{"className":45720,"style":3232},[2591],[507,45722],{},[507,45724],{"className":45725,"style":2715},[2714],[507,45727,2107],{"className":45728},[2719],[507,45730],{"className":45731,"style":2715},[2714],[507,45733,45735,45738,45742,45745,45748,45751,45754,45757,45760,45800],{"className":45734},[2561],[507,45736],{"className":45737,"style":2769},[2565],[507,45739,45741],{"className":45740},[2570],"ℏΩ",[507,45743],{"className":45744,"style":2965},[2714],[507,45746,37932],{"className":45747},[7570],[507,45749,580],{"className":45750},[2941],[507,45752,42346],{"className":45753,"style":2776},[2570,2611],[507,45755,3298],{"className":45756},[2570,2611],[507,45758,3649],{"className":45759},[2780],[507,45761,45763,45766],{"className":45762},[2570],[507,45764,44707],{"className":45765,"style":2776},[2570,2611],[507,45767,45769],{"className":45768},[2579],[507,45770,45772,45792],{"className":45771},[2583,3200],[507,45773,45775,45789],{"className":45774},[2587],[507,45776,45778],{"className":45777,"style":4507},[2591],[507,45779,45780,45783],{"style":4572},[507,45781],{"className":45782,"style":2600},[2599],[507,45784,45786],{"className":45785},[2604,2605,2606,2607],[507,45787,9139],{"className":45788},[2570,2611,2607],[507,45790,3225],{"className":45791},[3224],[507,45793,45795],{"className":45794},[2587],[507,45796,45798],{"className":45797,"style":3232},[2591],[507,45799],{},[507,45801,2819],{"className":45802},[2961],[18,45804,28454,45805,45865,45866,45894,45895,45932],{},[507,45806,45808,45829],{"className":45807},[2523],[507,45809,45811],{"className":45810},[2527],[2529,45812,45813],{"xmlns":2531},[2533,45814,45815,45827],{},[2536,45816,45817,45819,45821,45823,45825],{},[2542,45818,42346],{},[2689,45820,573],{},[2693,45822,584],{},[2542,45824,8563],{},[2542,45826,22278],{},[2549,45828,42607],{"encoding":2551},[507,45830,45832,45850],{"className":45831,"ariaHidden":2557},[2556],[507,45833,45835,45838,45841,45844,45847],{"className":45834},[2561],[507,45836],{"className":45837,"style":2639},[2565],[507,45839,42346],{"className":45840,"style":2776},[2570,2611],[507,45842],{"className":45843,"style":2919},[2714],[507,45845,573],{"className":45846},[2923],[507,45848],{"className":45849,"style":2919},[2714],[507,45851,45853,45856,45859,45862],{"className":45852},[2561],[507,45854],{"className":45855,"style":7035},[2565],[507,45857,584],{"className":45858},[2570],[507,45860,8563],{"className":45861,"style":2776},[2570,2611],[507,45863,22278],{"className":45864,"style":27338},[2570,2611]," is the drive angular frequency and ",[507,45867,45869,45882],{"className":45868},[2523],[507,45870,45872],{"className":45871},[2527],[2529,45873,45874],{"xmlns":2531},[2533,45875,45876,45880],{},[2536,45877,45878],{},[2542,45879,42662],{"mathvariant":2748},[2549,45881,42665],{"encoding":2551},[507,45883,45885],{"className":45884,"ariaHidden":2557},[2556],[507,45886,45888,45891],{"className":45887},[2561],[507,45889],{"className":45890,"style":2566},[2565],[507,45892,42662],{"className":45893},[2570]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[507,45896,45898,45917],{"className":45897},[2523],[507,45899,45901],{"className":45900},[2527],[2529,45902,45903],{"xmlns":2531},[2533,45904,45905,45915],{},[2536,45906,45907,45909,45911,45913],{},[2542,45908,42662],{"mathvariant":2748},[2542,45910,645],{"mathvariant":2748},[2693,45912,584],{},[2542,45914,8563],{},[2549,45916,42707],{"encoding":2551},[507,45918,45920],{"className":45919,"ariaHidden":2557},[2556],[507,45921,45923,45926,45929],{"className":45922},[2561],[507,45924],{"className":45925,"style":2769},[2565],[507,45927,42720],{"className":45928},[2570],[507,45930,8563],{"className":45931,"style":2776},[2570,2611]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[507,45934,45936],{"className":45935},[2784],[507,45937,45939,45998],{"className":45938},[2523],[507,45940,45942],{"className":45941},[2527],[2529,45943,45944],{"xmlns":2531,"display":2793},[2533,45945,45946,45995],{},[2536,45947,45948,45961,45963,45969,45993],{},[3168,45949,45950,45952],{},[2542,45951,3138],{},[2536,45953,45954,45956,45959],{},[2542,45955,20370],{"mathvariant":2748},[2542,45957,45958],{"mathvariant":2748},"W",[2542,45960,2658],{"mathvariant":2748},[2689,45962,573],{},[9491,45964,45965,45967],{},[2542,45966,43442],{"mathvariant":2748},[2693,45968,584],{},[2536,45970,45971,45973,45975,45981,45983,45985,45991],{},[2689,45972,580],{"fence":2557},[2542,45974,42745],{"mathvariant":2748},[3168,45976,45977,45979],{},[2542,45978,44707],{},[2542,45980,666],{},[2689,45982,2107],{},[2542,45984,42662],{"mathvariant":2748},[3168,45986,45987,45989],{},[2542,45988,44707],{},[2542,45990,9139],{},[2689,45992,3649],{"fence":2557},[2689,45994,2819],{"separator":2557},[2549,45996,45997],{"encoding":2551},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[507,45999,46001,46062],{"className":46000,"ariaHidden":2557},[2556],[507,46002,46004,46007,46053,46056,46059],{"className":46003},[2561],[507,46005],{"className":46006,"style":3187},[2565],[507,46008,46010,46013],{"className":46009},[2570],[507,46011,3138],{"className":46012,"style":3153},[2570,2611],[507,46014,46016],{"className":46015},[2579],[507,46017,46019,46045],{"className":46018},[2583,3200],[507,46020,46022,46042],{"className":46021},[2587],[507,46023,46025],{"className":46024,"style":3207},[2591],[507,46026,46027,46030],{"style":4510},[507,46028],{"className":46029,"style":2600},[2599],[507,46031,46033],{"className":46032},[2604,2605,2606,2607],[507,46034,46036],{"className":46035},[2570,2607],[507,46037,46039],{"className":46038},[2570,2607],[507,46040,43312],{"className":46041},[2570,43896,2607],[507,46043,3225],{"className":46044},[3224],[507,46046,46048],{"className":46047},[2587],[507,46049,46051],{"className":46050,"style":3232},[2591],[507,46052],{},[507,46054],{"className":46055,"style":2919},[2714],[507,46057,573],{"className":46058},[2923],[507,46060],{"className":46061,"style":2919},[2714],[507,46063,46065,46068,46130,46133,46237,46240],{"className":46064},[2561],[507,46066],{"className":46067,"style":45579},[2565],[507,46069,46071,46074,46127],{"className":46070},[2570],[507,46072],{"className":46073},[2941,9793],[507,46075,46077],{"className":46076},[9491],[507,46078,46080,46119],{"className":46079},[2583,3200],[507,46081,46083,46116],{"className":46082},[2587],[507,46084,46086,46097,46105],{"className":46085,"style":45598},[2591],[507,46087,46088,46091],{"style":11717},[507,46089],{"className":46090,"style":4310},[2599],[507,46092,46094],{"className":46093},[2570],[507,46095,584],{"className":46096},[2570],[507,46098,46099,46102],{"style":9878},[507,46100],{"className":46101,"style":4310},[2599],[507,46103],{"className":46104,"style":9886},[9885],[507,46106,46107,46110],{"style":9889},[507,46108],{"className":46109,"style":4310},[2599],[507,46111,46113],{"className":46112},[2570],[507,46114,43442],{"className":46115},[2570],[507,46117,3225],{"className":46118},[3224],[507,46120,46122],{"className":46121},[2587],[507,46123,46125],{"className":46124,"style":11755},[2591],[507,46126],{},[507,46128],{"className":46129},[2780,9793],[507,46131],{"className":46132,"style":2965},[2714],[507,46134,46136,46139,46142,46182,46185,46188,46191,46194,46234],{"className":46135},[2937],[507,46137,580],{"className":46138,"style":2943},[2941,2942],[507,46140,42745],{"className":46141},[2570],[507,46143,46145,46148],{"className":46144},[2570],[507,46146,44707],{"className":46147,"style":2776},[2570,2611],[507,46149,46151],{"className":46150},[2579],[507,46152,46154,46174],{"className":46153},[2583,3200],[507,46155,46157,46171],{"className":46156},[2587],[507,46158,46160],{"className":46159,"style":4507},[2591],[507,46161,46162,46165],{"style":4572},[507,46163],{"className":46164,"style":2600},[2599],[507,46166,46168],{"className":46167},[2604,2605,2606,2607],[507,46169,666],{"className":46170,"style":20582},[2570,2611,2607],[507,46172,3225],{"className":46173},[3224],[507,46175,46177],{"className":46176},[2587],[507,46178,46180],{"className":46179,"style":3232},[2591],[507,46181],{},[507,46183],{"className":46184,"style":2715},[2714],[507,46186,2107],{"className":46187},[2719],[507,46189],{"className":46190,"style":2715},[2714],[507,46192,42662],{"className":46193},[2570],[507,46195,46197,46200],{"className":46196},[2570],[507,46198,44707],{"className":46199,"style":2776},[2570,2611],[507,46201,46203],{"className":46202},[2579],[507,46204,46206,46226],{"className":46205},[2583,3200],[507,46207,46209,46223],{"className":46208},[2587],[507,46210,46212],{"className":46211,"style":4507},[2591],[507,46213,46214,46217],{"style":4572},[507,46215],{"className":46216,"style":2600},[2599],[507,46218,46220],{"className":46219},[2604,2605,2606,2607],[507,46221,9139],{"className":46222},[2570,2611,2607],[507,46224,3225],{"className":46225},[3224],[507,46227,46229],{"className":46228},[2587],[507,46230,46232],{"className":46231,"style":3232},[2591],[507,46233],{},[507,46235,3649],{"className":46236,"style":2943},[2780,2942],[507,46238],{"className":46239,"style":2965},[2714],[507,46241,2819],{"className":46242},[2961],[18,46244,46245],{},"with detuning",[507,46247,46249],{"className":46248},[2784],[507,46250,46252,46300],{"className":46251},[2523],[507,46253,46255],{"className":46254},[2527],[2529,46256,46257],{"xmlns":2531,"display":2793},[2533,46258,46259,46297],{},[2536,46260,46261,46263,46265,46267,46269,46275,46277,46279,46281,46283,46285,46287,46293,46295],{},[2542,46262,42745],{"mathvariant":2748},[2689,46264,573],{},[2542,46266,42346],{},[2689,46268,2691],{},[3168,46270,46271,46273],{},[2542,46272,42346],{},[2693,46274,601],{},[2689,46276,573],{},[2693,46278,584],{},[2542,46280,8563],{},[2689,46282,580],{"stretchy":2755},[2542,46284,22278],{},[2689,46286,2691],{},[3168,46288,46289,46291],{},[2542,46290,22278],{},[2693,46292,601],{},[2689,46294,3649],{"stretchy":2755},[2542,46296,53],{"mathvariant":2748},[2549,46298,46299],{"encoding":2551},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[507,46301,46303,46321,46339,46394,46421],{"className":46302,"ariaHidden":2557},[2556],[507,46304,46306,46309,46312,46315,46318],{"className":46305},[2561],[507,46307],{"className":46308,"style":2566},[2565],[507,46310,42745],{"className":46311},[2570],[507,46313],{"className":46314,"style":2919},[2714],[507,46316,573],{"className":46317},[2923],[507,46319],{"className":46320,"style":2919},[2714],[507,46322,46324,46327,46330,46333,46336],{"className":46323},[2561],[507,46325],{"className":46326,"style":2707},[2565],[507,46328,42346],{"className":46329,"style":2776},[2570,2611],[507,46331],{"className":46332,"style":2715},[2714],[507,46334,2691],{"className":46335},[2719],[507,46337],{"className":46338,"style":2715},[2714],[507,46340,46342,46345,46385,46388,46391],{"className":46341},[2561],[507,46343],{"className":46344,"style":25197},[2565],[507,46346,46348,46351],{"className":46347},[2570],[507,46349,42346],{"className":46350,"style":2776},[2570,2611],[507,46352,46354],{"className":46353},[2579],[507,46355,46357,46377],{"className":46356},[2583,3200],[507,46358,46360,46374],{"className":46359},[2587],[507,46361,46363],{"className":46362,"style":14281},[2591],[507,46364,46365,46368],{"style":4572},[507,46366],{"className":46367,"style":2600},[2599],[507,46369,46371],{"className":46370},[2604,2605,2606,2607],[507,46372,601],{"className":46373},[2570,2607],[507,46375,3225],{"className":46376},[3224],[507,46378,46380],{"className":46379},[2587],[507,46381,46383],{"className":46382,"style":3232},[2591],[507,46384],{},[507,46386],{"className":46387,"style":2919},[2714],[507,46389,573],{"className":46390},[2923],[507,46392],{"className":46393,"style":2919},[2714],[507,46395,46397,46400,46403,46406,46409,46412,46415,46418],{"className":46396},[2561],[507,46398],{"className":46399,"style":2769},[2565],[507,46401,584],{"className":46402},[2570],[507,46404,8563],{"className":46405,"style":2776},[2570,2611],[507,46407,580],{"className":46408},[2941],[507,46410,22278],{"className":46411,"style":27338},[2570,2611],[507,46413],{"className":46414,"style":2715},[2714],[507,46416,2691],{"className":46417},[2719],[507,46419],{"className":46420,"style":2715},[2714],[507,46422,46424,46427,46467,46470],{"className":46423},[2561],[507,46425],{"className":46426,"style":2769},[2565],[507,46428,46430,46433],{"className":46429},[2570],[507,46431,22278],{"className":46432,"style":27338},[2570,2611],[507,46434,46436],{"className":46435},[2579],[507,46437,46439,46459],{"className":46438},[2583,3200],[507,46440,46442,46456],{"className":46441},[2587],[507,46443,46445],{"className":46444,"style":14281},[2591],[507,46446,46447,46450],{"style":42267},[507,46448],{"className":46449,"style":2600},[2599],[507,46451,46453],{"className":46452},[2604,2605,2606,2607],[507,46454,601],{"className":46455},[2570,2607],[507,46457,3225],{"className":46458},[3224],[507,46460,46462],{"className":46461},[2587],[507,46463,46465],{"className":46464,"style":3232},[2591],[507,46466],{},[507,46468,3649],{"className":46469},[2780],[507,46471,53],{"className":46472},[2570],[18,46474,46475,46476,10799,46504,46532,46533,46561,46562,46590],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[507,46477,46479,46492],{"className":46478},[2523],[507,46480,46482],{"className":46481},[2527],[2529,46483,46484],{"xmlns":2531},[2533,46485,46486,46490],{},[2536,46487,46488],{},[2542,46489,42662],{"mathvariant":2748},[2549,46491,42665],{"encoding":2551},[507,46493,46495],{"className":46494,"ariaHidden":2557},[2556],[507,46496,46498,46501],{"className":46497},[2561],[507,46499],{"className":46500,"style":2566},[2565],[507,46502,42662],{"className":46503},[2570],[507,46505,46507,46520],{"className":46506},[2523],[507,46508,46510],{"className":46509},[2527],[2529,46511,46512],{"xmlns":2531},[2533,46513,46514,46518],{},[2536,46515,46516],{},[2542,46517,42745],{"mathvariant":2748},[2549,46519,42748],{"encoding":2551},[507,46521,46523],{"className":46522,"ariaHidden":2557},[2556],[507,46524,46526,46529],{"className":46525},[2561],[507,46527],{"className":46528,"style":2566},[2565],[507,46530,42745],{"className":46531},[2570],". The transverse component ",[507,46534,46536,46549],{"className":46535},[2523],[507,46537,46539],{"className":46538},[2527],[2529,46540,46541],{"xmlns":2531},[2533,46542,46543,46547],{},[2536,46544,46545],{},[2542,46546,42662],{"mathvariant":2748},[2549,46548,42665],{"encoding":2551},[507,46550,46552],{"className":46551,"ariaHidden":2557},[2556],[507,46553,46555,46558],{"className":46554},[2561],[507,46556],{"className":46557,"style":2566},[2565],[507,46559,42662],{"className":46560},[2570]," drives population transfer. A longitudinal component ",[507,46563,46565,46578],{"className":46564},[2523],[507,46566,46568],{"className":46567},[2527],[2529,46569,46570],{"xmlns":2531},[2533,46571,46572,46576],{},[2536,46573,46574],{},[2542,46575,42745],{"mathvariant":2748},[2549,46577,42748],{"encoding":2551},[507,46579,46581],{"className":46580,"ariaHidden":2557},[2556],[507,46582,46584,46587],{"className":46583},[2561],[507,46585],{"className":46586,"style":2566},[2565],[507,46588,42745],{"className":46589},[2570]," tilts the rotation axis away from the resonant direction.",[13,46592,46594],{"id":46593},"excited-state-probability","Excited-State Probability",[18,46596,46597,46598,46636,46637,46665],{},"The simulation assumes that the qubit begins in the ground state ",[507,46599,46601,46618],{"className":46600},[2523],[507,46602,46604],{"className":46603},[2527],[2529,46605,46606],{"xmlns":2531},[2533,46607,46608,46616],{},[2536,46609,46610,46612,46614],{},[2689,46611,2749],{"stretchy":2755},[2542,46613,37913],{},[2689,46615,2756],{"stretchy":2755},[2549,46617,41895],{"encoding":2551},[507,46619,46621],{"className":46620,"ariaHidden":2557},[2556],[507,46622,46624,46627,46630,46633],{"className":46623},[2561],[507,46625],{"className":46626,"style":2769},[2565],[507,46628,2749],{"className":46629},[2941],[507,46631,37913],{"className":46632,"style":2776},[2570,2611],[507,46634,2756],{"className":46635},[2780],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[507,46638,46640,46653],{"className":46639},[2523],[507,46641,46643],{"className":46642},[2527],[2529,46644,46645],{"xmlns":2531},[2533,46646,46647,46651],{},[2536,46648,46649],{},[2542,46650,41704],{},[2549,46652,41830],{"encoding":2551},[507,46654,46656],{"className":46655,"ariaHidden":2557},[2556],[507,46657,46659,46662],{"className":46658},[2561],[507,46660],{"className":46661,"style":2639},[2565],[507,46663,41704],{"className":46664,"style":41776},[2570,2611]," is",[507,46667,46669],{"className":46668},[2784],[507,46670,46672,46751],{"className":46671},[2523],[507,46673,46675],{"className":46674},[2527],[2529,46676,46677],{"xmlns":2531,"display":2793},[2533,46678,46679,46748],{},[2536,46680,46681,46687,46689,46691,46693,46695,46697,46699,46715,46726,46746],{},[3168,46682,46683,46685],{},[2542,46684,3174],{},[2542,46686,3286],{},[2689,46688,580],{"stretchy":2755},[2542,46690,22278],{},[2689,46692,2819],{"separator":2557},[2542,46694,41704],{},[2689,46696,3649],{"stretchy":2755},[2689,46698,573],{},[9491,46700,46701,46707],{},[2539,46702,46703,46705],{},[2542,46704,42662],{"mathvariant":2748},[2693,46706,584],{},[3775,46708,46709,46711,46713],{},[2542,46710,42662],{"mathvariant":2748},[2542,46712,20370],{},[2693,46714,584],{},[2539,46716,46717,46724],{},[2536,46718,46719,46722],{},[2542,46720,46721],{},"sin",[2689,46723,36690],{},[2693,46725,584],{},[2536,46727,46728,46730,46744],{},[2689,46729,580],{"fence":2557},[9491,46731,46732,46742],{},[2536,46733,46734,46740],{},[3168,46735,46736,46738],{},[2542,46737,42662],{"mathvariant":2748},[2542,46739,20370],{},[2542,46741,41704],{},[2693,46743,584],{},[2689,46745,3649],{"fence":2557},[2689,46747,2819],{"separator":2557},[2549,46749,46750],{"encoding":2551},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right),",[507,46752,46754,46827],{"className":46753,"ariaHidden":2557},[2556],[507,46755,46757,46760,46800,46803,46806,46809,46812,46815,46818,46821,46824],{"className":46756},[2561],[507,46758],{"className":46759,"style":2769},[2565],[507,46761,46763,46766],{"className":46762},[2570],[507,46764,3174],{"className":46765,"style":3220},[2570,2611],[507,46767,46769],{"className":46768},[2579],[507,46770,46772,46792],{"className":46771},[2583,3200],[507,46773,46775,46789],{"className":46774},[2587],[507,46776,46778],{"className":46777,"style":4507},[2591],[507,46779,46780,46783],{"style":7637},[507,46781],{"className":46782,"style":2600},[2599],[507,46784,46786],{"className":46785},[2604,2605,2606,2607],[507,46787,3286],{"className":46788},[2570,2611,2607],[507,46790,3225],{"className":46791},[3224],[507,46793,46795],{"className":46794},[2587],[507,46796,46798],{"className":46797,"style":3232},[2591],[507,46799],{},[507,46801,580],{"className":46802},[2941],[507,46804,22278],{"className":46805,"style":27338},[2570,2611],[507,46807,2819],{"className":46808},[2961],[507,46810],{"className":46811,"style":2965},[2714],[507,46813,41704],{"className":46814,"style":41776},[2570,2611],[507,46816,3649],{"className":46817},[2780],[507,46819],{"className":46820,"style":2919},[2714],[507,46822,573],{"className":46823},[2923],[507,46825],{"className":46826,"style":2919},[2714],[507,46828,46830,46834,46976,46979,47010,47013,47131,47134],{"className":46829},[2561],[507,46831],{"className":46832,"style":46833},[2565],"height:2.4706em;vertical-align:-0.9795em;",[507,46835,46837,46840,46973],{"className":46836},[2570],[507,46838],{"className":46839},[2941,9793],[507,46841,46843],{"className":46842},[9491],[507,46844,46846,46964],{"className":46845},[2583,3200],[507,46847,46849,46961],{"className":46848},[2587],[507,46850,46853,46916,46924],{"className":46851,"style":46852},[2591],"height:1.4911em;",[507,46854,46855,46858],{"style":11717},[507,46856],{"className":46857,"style":4310},[2599],[507,46859,46861],{"className":46860},[2570],[507,46862,46864,46867],{"className":46863},[2570],[507,46865,42662],{"className":46866},[2570],[507,46868,46870],{"className":46869},[2579],[507,46871,46873,46907],{"className":46872},[2583,3200],[507,46874,46876,46904],{"className":46875},[2587],[507,46877,46880,46892],{"className":46878,"style":46879},[2591],"height:0.7959em;",[507,46881,46883,46886],{"style":46882},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[507,46884],{"className":46885,"style":2600},[2599],[507,46887,46889],{"className":46888},[2604,2605,2606,2607],[507,46890,20370],{"className":46891,"style":20395},[2570,2611,2607],[507,46893,46895,46898],{"style":46894},"top:-3.0448em;margin-right:0.05em;",[507,46896],{"className":46897,"style":2600},[2599],[507,46899,46901],{"className":46900},[2604,2605,2606,2607],[507,46902,584],{"className":46903},[2570,2607],[507,46905,3225],{"className":46906},[3224],[507,46908,46910],{"className":46909},[2587],[507,46911,46914],{"className":46912,"style":46913},[2591],"height:0.2935em;",[507,46915],{},[507,46917,46918,46921],{"style":9878},[507,46919],{"className":46920,"style":4310},[2599],[507,46922],{"className":46923,"style":9886},[9885],[507,46925,46926,46929],{"style":9889},[507,46927],{"className":46928,"style":4310},[2599],[507,46930,46932],{"className":46931},[2570],[507,46933,46935,46938],{"className":46934},[2570],[507,46936,42662],{"className":46937},[2570],[507,46939,46941],{"className":46940},[2579],[507,46942,46944],{"className":46943},[2583],[507,46945,46947],{"className":46946},[2587],[507,46948,46950],{"className":46949,"style":13224},[2591],[507,4695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561],[507,47246],{"className":47247,"style":47248},[2565],"height:1.24em;vertical-align:-0.1777em;",[507,47250,47252],{"className":47251},[2570,9819],[507,47253,47255,47361],{"className":47254},[2583,3200],[507,47256,47258,47358],{"className":47257},[2587],[507,47259,47262,47341],{"className":47260,"style":47261},[2591],"height:1.0623em;",[507,47263,47266,47270],{"className":47264,"style":47265},[9833],"top:-3.2em;",[507,47267],{"className":47268,"style":47269},[2599],"height:3.2em;",[507,47271,47274,47303,47306,47309,47312],{"className":47272,"style":47273},[2570],"padding-left:1em;",[507,47275,47277,47280],{"className":47276},[2570],[507,47278,42662],{"className":47279},[2570],[507,47281,47283],{"className":47282},[2579],[507,47284,47286],{"className":47285},[2583],[507,47287,47289],{"className":47288},[2587],[507,47290,47292],{"className":47291,"style":29409},[2591],[507,47293,47294,47297],{"style":29412},[507,47295],{"className":47296,"style":2600},[2599],[507,47298,47300],{"className":47299},[2604,2605,2606,2607],[507,47301,584],{"className":47302},[2570,2607],[507,47304],{"className":47305,"style":2715},[2714],[507,47307,2107],{"className":47308},[2719],[507,47310],{"className":47311,"style":2715},[2714],[507,47313,47315,47318],{"className":47314},[2570],[507,47316,42745],{"className":47317},[2570],[507,47319,47321],{"className":47320},[2579],[507,47322,47324],{"className":47323},[2583],[507,47325,47327],{"className":47326},[2587],[507,47328,47330],{"className":47329,"style":29409},[2591],[507,47331,47332,47335],{"style":29412},[507,47333],{"className":47334,"style":2600},[2599],[507,47336,47338],{"className":47337},[2604,2605,2606,2607],[507,47339,584],{"className":47340},[2570,2607],[507,47342,47344,47347],{"style":47343},"top:-3.0223em;",[507,47345],{"className":47346,"style":47269},[2599],[507,47348,47351],{"className":47349,"style":47350},[9853],"min-width:1.02em;height:1.28em;",[6281,47352,47355],{"xmlns":6283,"width":9857,"height":47353,"viewBox":47354,"preserveAspectRatio":9860},"1.28em","0 0 400000 1296",[6290,47356],{"d":47357},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[507,47359,3225],{"className":47360},[3224],[507,47362,47364],{"className":47363},[2587],[507,47365,47368],{"className":47366,"style":47367},[2591],"height:0.1777em;",[507,47369],{},[18,47371,47372],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[507,47374,47376],{"className":47375},[2784],[507,47377,47379,47407],{"className":47378},[2523],[507,47380,47382],{"className":47381},[2527],[2529,47383,47384],{"xmlns":2531,"display":2793},[2533,47385,47386,47404],{},[2536,47387,47388],{},[9491,47389,47390,47396],{},[2539,47391,47392,47394],{},[2542,47393,42662],{"mathvariant":2748},[2693,47395,584],{},[3775,47397,47398,47400,47402],{},[2542,47399,42662],{"mathvariant":2748},[2542,47401,20370],{},[2693,47403,584],{},[2549,47405,47406],{"encoding":2551},"\\frac{\\Omega^2}{\\Omega_R^2}",[507,47408,47410],{"className":47409,"ariaHidden":2557},[2556],[507,47411,47413,47416],{"className":47412},[2561],[507,47414],{"className":47415,"style":46833},[2565],[507,47417,47419,47422,47549],{"className":47418},[2570],[507,47420],{"className":47421},[2941,9793],[507,47423,47425],{"className":47424},[9491],[507,47426,47428,47541],{"className":47427},[2583,3200],[507,47429,47431,47538],{"className":47430},[2587],[507,47432,47434,47493,47501],{"className":47433,"style":46852},[2591],[507,47435,47436,47439],{"style":11717},[507,47437],{"className":47438,"style":4310},[2599],[507,47440,47442],{"className":47441},[2570],[507,47443,47445,47448],{"className":47444},[2570],[507,47446,42662],{"className":47447},[2570],[507,47449,47451],{"className":47450},[2579],[507,47452,47454,47485],{"className":47453},[2583,3200],[507,47455,47457,47482],{"className":47456},[2587],[507,47458,47460,47471],{"className":47459,"style":46879},[2591],[507,47461,47462,47465],{"style":46882},[507,47463],{"className":47464,"style":2600},[2599],[507,47466,47468],{"className":47467},[2604,2605,2606,2607],[507,47469,20370],{"className":47470,"style":20395},[2570,2611,2607],[507,47472,47473,47476],{"style":46894},[507,47474],{"className":47475,"style":2600},[2599],[507,47477,47479],{"className":47478},[2604,2605,2606,2607],[507,47480,584],{"className":47481},[2570,2607],[507,47483,3225],{"className":47484},[3224],[507,47486,47488],{"className":47487},[2587],[507,47489,47491],{"className":47490,"style":46913},[2591],[507,47492],{},[507,47494,47495,47498],{"style":9878},[507,47496],{"className":47497,"style":4310},[2599],[507,47499],{"className":47500,"style":9886},[9885],[507,47502,47503,47506],{"style":9889},[507,47504],{"className":47505,"style":4310},[2599],[507,47507,47509],{"className":47508},[2570],[507,47510,47512,47515],{"className":47511},[2570],[507,47513,42662],{"className":47514},[2570],[507,47516,47518],{"className":47517},[2579],[507,47519,47521],{"className":47520},[2583],[507,47522,47524],{"className":47523},[2587],[507,47525,47527],{"className":47526,"style":13224},[2591],[507,47528,47529,47532],{"style":2595},[507,47530],{"className":47531,"style":2600},[2599],[507,47533,47535],{"className":47534},[2604,2605,2606,2607],[507,47536,584],{"className":47537},[2570,2607],[507,47539,3225],{"className":47540},[3224],[507,47542,47544],{"className":47543},[2587],[507,47545,47547],{"className":47546,"style":46970},[2591],[507,47548],{},[507,47550],{"className":47551},[2780,9793],[18,47553,47554,47555,47606],{},"sets the maximum possible population transfer. At exact resonance, ",[507,47556,47558,47576],{"className":47557},[2523],[507,47559,47561],{"className":47560},[2527],[2529,47562,47563],{"xmlns":2531},[2533,47564,47565,47573],{},[2536,47566,47567,47569,47571],{},[2542,47568,42745],{"mathvariant":2748},[2689,47570,573],{},[2693,47572,601],{},[2549,47574,47575],{"encoding":2551},"\\Delta=0",[507,47577,47579,47597],{"className":47578,"ariaHidden":2557},[2556],[507,47580,47582,47585,47588,47591,47594],{"className":47581},[2561],[507,47583],{"className":47584,"style":2566},[2565],[507,47586,42745],{"className":47587},[2570],[507,47589],{"className":47590,"style":2919},[2714],[507,47592,573],{"className":47593},[2923],[507,47595],{"className":47596,"style":2919},[2714],[507,47598,47600,47603],{"className":47599},[2561],[507,47601],{"className":47602,"style":2729},[2565],[507,47604,601],{"className":47605},[2570],", so",[507,47608,47610],{"className":47609},[2784],[507,47611,47613,47645],{"className":47612},[2523],[507,47614,47616],{"className":47615},[2527],[2529,47617,47618],{"xmlns":2531,"display":2793},[2533,47619,47620,47642],{},[2536,47621,47622,47638,47640],{},[9491,47623,47624,47630],{},[2539,47625,47626,47628],{},[2542,47627,42662],{"mathvariant":2748},[2693,47629,584],{},[3775,47631,47632,47634,47636],{},[2542,47633,42662],{"mathvariant":2748},[2542,47635,20370],{},[2693,47637,584],{},[2689,47639,573],{},[2693,47641,44343],{},[2549,47643,47644],{"encoding":2551},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[507,47646,47648,47799],{"className":47647,"ariaHidden":2557},[2556],[507,47649,47651,47654,47790,47793,47796],{"className":47650},[2561],[507,47652],{"className":47653,"style":46833},[2565],[507,47655,47657,47660,47787],{"className":47656},[2570],[507,47658],{"className":47659},[2941,9793],[507,47661,47663],{"className":47662},[9491],[507,47664,47666,47779],{"className":47665},[2583,3200],[507,47667,47669,47776],{"className":47668},[2587],[507,47670,47672,47731,47739],{"className":47671,"style":46852},[2591],[507,47673,47674,47677],{"style":11717},[507,47675],{"className":47676,"style":4310},[2599],[507,47678,47680],{"className":47679},[2570],[507,47681,47683,47686],{"className":47682},[2570],[507,47684,42662],{"className":47685},[2570],[507,47687,47689],{"className":47688},[2579],[507,47690,47692,47723],{"className":47691},[2583,3200],[507,47693,47695,47720],{"className":47694},[2587],[507,47696,47698,47709],{"className":47697,"style":46879},[2591],[507,47699,47700,47703],{"style":46882},[507,47701],{"className":47702,"style":2600},[2599],[507,47704,47706],{"className":47705},[2604,2605,2606,2607],[507,47707,20370],{"className":47708,"style":20395},[2570,2611,2607],[507,47710,47711,47714],{"style":46894},[507,47712],{"className":47713,"style":2600},[2599],[507,47715,47717],{"className":47716},[2604,2605,2606,2607],[507,47718,584],{"className":47719},[2570,2607],[507,47721,3225],{"className":47722},[3224],[507,47724,47726],{"className":47725},[2587],[507,47727,47729],{"className":47728,"style":46913},[2591],[507,47730],{},[507,47732,47733,47736],{"style":9878},[507,47734],{"className":47735,"style":4310},[2599],[507,47737],{"className":47738,"style":9886},[9885],[507,47740,47741,47744],{"style":9889},[507,47742],{"className":47743,"style":4310},[2599],[507,47745,47747],{"className":47746},[2570],[507,47748,47750,47753],{"className":47749},[2570],[507,47751,42662],{"className":47752},[2570],[507,47754,47756],{"className":47755},[2579],[507,47757,47759],{"className":47758},[2583],[507,47760,47762],{"className":47761},[2587],[507,47763,47765],{"className":47764,"style":13224},[2591],[507,47766,47767,47770],{"style":2595},[507,47768],{"className":47769,"style":2600},[2599],[507,47771,47773],{"className":47772},[2604,2605,2606,2607],[507,47774,584],{"className":47775},[2570,2607],[507,47777,3225],{"className":47778},[3224],[507,47780,47782],{"className":47781},[2587],[507,47783,47785],{"className":47784,"style":46970},[2591],[507,47786],{},[507,47788],{"className":47789},[2780,9793],[507,47791],{"className":47792,"style":2919},[2714],[507,47794,573],{"className":47795},[2923],[507,47797],{"className":47798,"style":2919},[2714],[507,47800,47802,47805],{"className":47801},[2561],[507,47803],{"className":47804,"style":2729},[2565],[507,47806,44343],{"className":47807},[2570],[18,47809,47810],{},"Full population inversion is achievable in the ideal resonant model. For any nonzero detuning, the coefficient falls below one, so the state generally cannot reach unit excited-state probability.",[18,47812,47813],{},"At resonance, the excited-state probability simplifies to",[507,47815,47817],{"className":47816},[2784],[507,47818,47820,47874],{"className":47819},[2523],[507,47821,47823],{"className":47822},[2527],[2529,47824,47825],{"xmlns":2531,"display":2793},[2533,47826,47827,47871],{},[2536,47828,47829,47835,47837,47839,47841,47843,47853,47869],{},[3168,47830,47831,47833],{},[2542,47832,3174],{},[2542,47834,3286],{},[2689,47836,580],{"stretchy":2755},[2542,47838,41704],{},[2689,47840,3649],{"stretchy":2755},[2689,47842,573],{},[2539,47844,47845,47851],{},[2536,47846,47847,47849],{},[2542,47848,46721],{},[2689,47850,36690],{},[2693,47852,584],{},[2536,47854,47855,47857,47867],{},[2689,47856,580],{"fence":2557},[9491,47858,47859,47865],{},[2536,47860,47861,47863],{},[2542,47862,42662],{"mathvariant":2748},[2542,47864,41704],{},[2693,47866,584],{},[2689,47868,3649],{"fence":2557},[2542,47870,53],{"mathvariant":2748},[2549,47872,47873],{"encoding":2551},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[507,47875,47877,47941],{"className":47876,"ariaHidden":2557},[2556],[507,47878,47880,47883,47923,47926,47929,47932,47935,47938],{"className":47879},[2561],[507,47881],{"className":47882,"style":2769},[2565],[507,47884,47886,47889],{"className":47885},[2570],[507,47887,3174],{"className":47888,"style":3220},[2570,2611],[507,47890,47892],{"className":47891},[2579],[507,47893,47895,47915],{"className":47894},[2583,3200],[507,47896,47898,47912],{"className":47897},[2587],[507,47899,47901],{"className":47900,"style":4507},[2591],[507,47902,47903,47906],{"style":7637},[507,47904],{"className":47905,"style":2600},[2599],[507,47907,47909],{"className":47908},[2604,2605,2606,2607],[507,47910,3286],{"className":47911},[2570,2611,2607],[507,47913,3225],{"className":47914},[3224],[507,47916,47918],{"className":47917},[2587],[507,47919,47921],{"className":47920,"style":3232},[2591],[507,47922],{},[507,47924,580],{"className":47925},[2941],[507,47927,41704],{"className":47928,"style":41776},[2570,2611],[507,47930,3649],{"className":47931},[2780],[507,47933],{"className":47934,"style":2919},[2714],[507,47936,573],{"className":47937},[2923],[507,47939],{"className":47940,"style":2919},[2714],[507,47942,47944,47947,47976,47979,48059,48062],{"className":47943},[2561],[507,47945],{"className":47946,"style":37668},[2565],[507,47948,47950,47953],{"className":47949},[7570],[507,47951,46721],{"className":47952},[7570],[507,47954,47956],{"className":47955},[2579],[507,47957,47959],{"className":47958},[2583],[507,47960,47962],{"className":47961},[2587],[507,47963,47965],{"className":47964,"style":46997},[2591],[507,47966,47967,47970],{"style":47000},[507,47968],{"className":47969,"style":2600},[2599],[507,47971,47973],{"className":47972},[2604,2605,2606,2607],[507,47974,584],{"className":47975},[2570,2607],[507,47977],{"className":47978,"style":2965},[2714],[507,47980,47982,47988,48053],{"className":47981},[2937],[507,47983,47985],{"className":47984,"style":2943},[2941,2942],[507,47986,580],{"className":47987},[2947,2606],[507,47989,47991,47994,48050],{"className":47990},[2570],[507,47992],{"className":47993},[2941,9793],[507,47995,47997],{"className":47996},[9491],[507,47998,48000,48042],{"className":47999},[2583,3200],[507,48001,48003,48039],{"className":48002},[2587],[507,48004,48006,48017,48025],{"className":48005,"style":47040},[2591],[507,48007,48008,48011],{"style":11717},[507,48009],{"className":48010,"style":4310},[2599],[507,48012,48014],{"className":48013},[2570],[507,48015,584],{"className":48016},[2570],[507,48018,48019,48022],{"style":9878},[507,48020],{"className":48021,"style":4310},[2599],[507,48023],{"className":48024,"style":9886},[9885],[507,48026,48027,48030],{"style":9889},[507,48028],{"className":48029,"style":4310},[2599],[507,48031,48033,48036],{"className":48032},[2570],[507,48034,42662],{"className":48035},[2570],[507,48037,41704],{"className":48038,"style":41776},[2570,2611],[507,48040,3225],{"className":48041},[3224],[507,48043,48045],{"className":48044},[2587],[507,48046,48048],{"className":48047,"style":11755},[2591],[507,48049],{},[507,48051],{"className":48052},[2780,9793],[507,48054,48056],{"className":48055,"style":2943},[2780,2942],[507,48057,3649],{"className":48058},[2947,2606],[507,48060],{"className":48061,"style":2965},[2714],[507,48063,53],{"className":48064},[2570],[18,48066,48067,48068,48097],{},"The pulse duration required for ideal population inversion defines the standard ",[507,48069,48071,48085],{"className":48070},[2523],[507,48072,48074],{"className":48073},[2527],[2529,48075,48076],{"xmlns":2531},[2533,48077,48078,48082],{},[2536,48079,48080],{},[2542,48081,8563],{},[2549,48083,48084],{"encoding":2551},"\\pi",[507,48086,48088],{"className":48087,"ariaHidden":2557},[2556],[507,48089,48091,48094],{"className":48090},[2561],[507,48092],{"className":48093,"style":2639},[2565],[507,48095,8563],{"className":48096,"style":2776},[2570,2611],"-pulse time,",[507,48099,48101],{"className":48100},[2784],[507,48102,48104,48154],{"className":48103},[2523],[507,48105,48107],{"className":48106},[2527],[2529,48108,48109],{"xmlns":2531,"display":2793},[2533,48110,48111,48151],{},[2536,48112,48113,48119,48121,48127,48129,48149],{},[3168,48114,48115,48117],{},[2542,48116,41704],{},[2542,48118,8563],{},[2689,48120,573],{},[9491,48122,48123,48125],{},[2542,48124,8563],{},[2542,48126,42662],{"mathvariant":2748},[2689,48128,573],{},[9491,48130,48131,48133],{},[2693,48132,625],{},[2536,48134,48135,48137,48139,48141,48143,48145,48147],{},[2693,48136,584],{},[2689,48138,580],{"stretchy":2755},[2542,48140,42662],{"mathvariant":2748},[2542,48142,645],{"mathvariant":2748},[2693,48144,584],{},[2542,48146,8563],{},[2689,48148,3649],{"stretchy":2755},[2542,48150,53],{"mathvariant":2748},[2549,48152,48153],{"encoding":2551},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[507,48155,48157,48213,48292],{"className":48156,"ariaHidden":2557},[2556],[507,48158,48160,48163,48204,48207,48210],{"className":48159},[2561],[507,48161],{"className":48162,"style":25197},[2565],[507,48164,48166,48169],{"className":48165},[2570],[507,48167,41704],{"className":48168,"style":41776},[2570,2611],[507,48170,48172],{"className":48171},[2579],[507,48173,48175,48196],{"className":48174},[2583,3200],[507,48176,48178,48193],{"className":48177},[2587],[507,48179,48181],{"className":48180,"style":4507},[2591],[507,48182,48184,48187],{"style":48183},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[507,48185],{"className":48186,"style":2600},[2599],[507,48188,48190],{"className":48189},[2604,2605,2606,2607],[507,48191,8563],{"className":48192,"style":2776},[2570,2611,2607],[507,48194,3225],{"className":48195},[3224],[507,48197,48199],{"className":48198},[2587],[507,48200,48202],{"className":48201,"style":3232},[2591],[507,48203],{},[507,48205],{"className":48206,"style":2919},[2714],[507,48208,573],{"className":48209},[2923],[507,48211],{"className":48212,"style":2919},[2714],[507,48214,48216,48220,48283,48286,48289],{"className":48215},[2561],[507,48217],{"className":48218,"style":48219},[2565],"height:1.7936em;vertical-align:-0.686em;",[507,48221,48223,48226,48280],{"className":48222},[2570],[507,48224],{"className":48225},[2941,9793],[507,48227,48229],{"className":48228},[9491],[507,48230,48232,48272],{"className":48231},[2583,3200],[507,48233,48235,48269],{"className":48234},[2587],[507,48236,48239,48250,48258],{"className":48237,"style":48238},[2591],"height:1.1076em;",[507,48240,48241,48244],{"style":11717},[507,48242],{"className":48243,"style":4310},[2599],[507,48245,48247],{"className":48246},[2570],[507,48248,42662],{"className":48249},[2570],[507,48251,48252,48255],{"style":9878},[507,48253],{"className":48254,"style":4310},[2599],[507,48256],{"className":48257,"style":9886},[9885],[507,48259,48260,48263],{"style":9889},[507,48261],{"className":48262,"style":4310},[2599],[507,48264,48266],{"className":48265},[2570],[507,48267,8563],{"className":48268,"style":2776},[2570,2611],[507,48270,3225],{"className":48271},[3224],[507,48273,48275],{"className":48274},[2587],[507,48276,48278],{"className":48277,"style":11755},[2591],[507,48279],{},[507,48281],{"className":48282},[2780,9793],[507,48284],{"className":48285,"style":2919},[2714],[507,48287,573],{"className":48288},[2923],[507,48290],{"className":48291,"style":2919},[2714],[507,48293,48295,48298,48372],{"className":48294},[2561],[507,48296],{"className":48297,"style":35838},[2565],[507,48299,48301,48304,48369],{"className":48300},[2570],[507,48302],{"className":48303},[2941,9793],[507,48305,48307],{"className":48306},[9491],[507,48308,48310,48361],{"className":48309},[2583,3200],[507,48311,48313,48358],{"className":48312},[2587],[507,48314,48316,48339,48347],{"className":48315,"style":9806},[2591],[507,48317,48318,48321],{"style":11717},[507,48319],{"className":48320,"style":4310},[2599],[507,48322,48324,48327,48330,48333,48336],{"className":48323},[2570],[507,48325,584],{"className":48326},[2570],[507,48328,580],{"className":48329},[2941],[507,48331,42720],{"className":48332},[2570],[507,48334,8563],{"className":48335,"style":2776},[2570,2611],[507,48337,3649],{"className":48338},[2780],[507,48340,48341,48344],{"style":9878},[507,48342],{"className":48343,"style":4310},[2599],[507,48345],{"className":48346,"style":9886},[9885],[507,48348,48349,48352],{"style":9889},[507,48350],{"className":48351,"style":4310},[2599],[507,48353,48355],{"className":48354},[2570],[507,48356,625],{"className":48357},[2570],[507,48359,3225],{"className":48360},[3224],[507,48362,48364],{"className":48363},[2587],[507,48365,48367],{"className":48366,"style":29556},[2591],[507,48368],{},[507,48370],{"className":48371},[2780,9793],[507,48373,53],{"className":48374},[2570],[18,48376,48377],{},"Given the default drive strength",[507,48379,48381],{"className":48380},[2784],[507,48382,48384,48424],{"className":48383},[2523],[507,48385,48387],{"className":48386},[2527],[2529,48388,48389],{"xmlns":2531,"display":2793},[2533,48390,48391,48421],{},[2536,48392,48393,48403,48405,48408,48410,48419],{},[9491,48394,48395,48397],{},[2542,48396,42662],{"mathvariant":2748},[2536,48398,48399,48401],{},[2693,48400,584],{},[2542,48402,8563],{},[2689,48404,573],{},[2693,48406,48407],{},"20",[6167,48409,6169],{},[2536,48411,48412,48415,48417],{},[2542,48413,48414],{"mathvariant":2748},"M",[2542,48416,3138],{"mathvariant":2748},[2542,48418,666],{"mathvariant":2748},[2689,48420,2819],{"separator":2557},[2549,48422,48423],{"encoding":2551},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[507,48425,48427,48508],{"className":48426,"ariaHidden":2557},[2556],[507,48428,48430,48434,48499,48502,48505],{"className":48429},[2561],[507,48431],{"className":48432,"style":48433},[2565],"height:2.0463em;vertical-align:-0.686em;",[507,48435,48437,48440,48496],{"className":48436},[2570],[507,48438],{"className":48439},[2941,9793],[507,48441,48443],{"className":48442},[9491],[507,48444,48446,48488],{"className":48445},[2583,3200],[507,48447,48449,48485],{"className":48448},[2587],[507,48450,48452,48466,48474],{"className":48451,"style":47040},[2591],[507,48453,48454,48457],{"style":11717},[507,48455],{"className":48456,"style":4310},[2599],[507,48458,48460,48463],{"className":48459},[2570],[507,48461,584],{"className":48462},[2570],[507,48464,8563],{"className":48465,"style":2776},[2570,2611],[507,48467,48468,48471],{"style":9878},[507,48469],{"className":48470,"style":4310},[2599],[507,48472],{"className":48473,"style":9886},[9885],[507,48475,48476,48479],{"style":9889},[507,48477],{"className":48478,"style":4310},[2599],[507,48480,48482],{"className":48481},[2570],[507,48483,42662],{"className":48484},[2570],[507,48486,3225],{"className":48487},[3224],[507,48489,48491],{"className":48490},[2587],[507,48492,48494],{"className":48493,"style":11755},[2591],[507,48495],{},[507,48497],{"className":48498},[2780,9793],[507,48500],{"className":48501,"style":2919},[2714],[507,48503,573],{"className":48504},[2923],[507,48506],{"className":48507,"style":2919},[2714],[507,48509,48511,48515,48518,48521,48528],{"className":48510},[2561],[507,48512],{"className":48513,"style":48514},[2565],"height:0.8778em;vertical-align:-0.1944em;",[507,48516,48407],{"className":48517},[2570],[507,48519,6169],{"className":48520},[2714,43889],[507,48522,48524],{"className":48523},[2570],[507,48525,48527],{"className":48526},[2570,43896],"MHz",[507,48529,2819],{"className":48530},[2961],[18,48532,48533,48534,48562],{},"the ideal ",[507,48535,48537,48550],{"className":48536},[2523],[507,48538,48540],{"className":48539},[2527],[2529,48541,48542],{"xmlns":2531},[2533,48543,48544,48548],{},[2536,48545,48546],{},[2542,48547,8563],{},[2549,48549,48084],{"encoding":2551},[507,48551,48553],{"className":48552,"ariaHidden":2557},[2556],[507,48554,48556,48559],{"className":48555},[2561],[507,48557],{"className":48558,"style":2639},[2565],[507,48560,8563],{"className":48561,"style":2776},[2570,2611],"-pulse time is",[507,48564,48566],{"className":48565},[2784],[507,48567,48569,48602],{"className":48568},[2523],[507,48570,48572],{"className":48571},[2527],[2529,48573,48574],{"xmlns":2531,"display":2793},[2533,48575,48576,48599],{},[2536,48577,48578,48584,48586,48589,48591,48597],{},[3168,48579,48580,48582],{},[2542,48581,41704],{},[2542,48583,8563],{},[2689,48585,573],{},[2693,48587,48588],{},"25",[6167,48590,6169],{},[2536,48592,48593,48595],{},[2542,48594,4420],{"mathvariant":2748},[2542,48596,14533],{"mathvariant":2748},[2542,48598,53],{"mathvariant":2748},[2549,48600,48601],{"encoding":2551},"\\tau_\\pi=25~\\mathrm{ns}.",[507,48603,48605,48660],{"className":48604,"ariaHidden":2557},[2556],[507,48606,48608,48611,48651,48654,48657],{"className":48607},[2561],[507,48609],{"className":48610,"style":25197},[2565],[507,48612,48614,48617],{"className":48613},[2570],[507,48615,41704],{"className":48616,"style":41776},[2570,2611],[507,48618,48620],{"className":48619},[2579],[507,48621,48623,48643],{"className":48622},[2583,3200],[507,48624,48626,48640],{"className":48625},[2587],[507,48627,48629],{"className":48628,"style":4507},[2591],[507,48630,48631,48634],{"style":48183},[507,48632],{"className":48633,"style":2600},[2599],[507,48635,48637],{"className":48636},[2604,2605,2606,2607],[507,48638,8563],{"className":48639,"style":2776},[2570,2611,2607],[507,48641,3225],{"className":48642},[3224],[507,48644,48646],{"className":48645},[2587],[507,48647,48649],{"className":48648,"style":3232},[2591],[507,48650],{},[507,48652],{"className":48653,"style":2919},[2714],[507,48655,573],{"className":48656},[2923],[507,48658],{"className":48659,"style":2919},[2714],[507,48661,48663,48666,48669,48672,48679],{"className":48662},[2561],[507,48664],{"className":48665,"style":2729},[2565],[507,48667,48588],{"className":48668},[2570],[507,48670,6169],{"className":48671},[2714,43889],[507,48673,48675],{"className":48674},[2570],[507,48676,48678],{"className":48677},[2570,43896],"ns",[507,48680,53],{"className":48681},[2570],[18,48683,48684],{},"A full period of the resonant excited-state probability oscillation is",[507,48686,48688],{"className":48687},[2784],[507,48689,48691,48739],{"className":48690},[2523],[507,48692,48694],{"className":48693},[2527],[2529,48695,48696],{"xmlns":2531,"display":2793},[2533,48697,48698,48736],{},[2536,48699,48700,48706,48708,48718,48720,48734],{},[3168,48701,48702,48704],{},[2542,48703,37046],{},[2542,48705,20370],{},[2689,48707,573],{},[9491,48709,48710,48716],{},[2536,48711,48712,48714],{},[2693,48713,584],{},[2542,48715,8563],{},[2542,48717,42662],{"mathvariant":2748},[2689,48719,573],{},[9491,48721,48722,48724],{},[2693,48723,625],{},[2536,48725,48726,48728,48730,48732],{},[2542,48727,42662],{"mathvariant":2748},[2542,48729,645],{"mathvariant":2748},[2693,48731,584],{},[2542,48733,8563],{},[2542,48735,53],{"mathvariant":2748},[2549,48737,48738],{"encoding":2551},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[507,48740,48742,48797,48878],{"className":48741,"ariaHidden":2557},[2556],[507,48743,48745,48748,48788,48791,48794],{"className":48744},[2561],[507,48746],{"className":48747,"style":3187},[2565],[507,48749,48751,48754],{"className":48750},[2570],[507,48752,37046],{"className":48753,"style":3220},[2570,2611],[507,48755,48757],{"className":48756},[2579],[507,48758,48760,48780],{"className":48759},[2583,3200],[507,48761,48763,48777],{"className":48762},[2587],[507,48764,48766],{"className":48765,"style":3207},[2591],[507,48767,48768,48771],{"style":7637},[507,48769],{"className":48770,"style":2600},[2599],[507,48772,48774],{"className":48773},[2604,2605,2606,2607],[507,48775,20370],{"className":48776,"style":20395},[2570,2611,2607],[507,48778,3225],{"className":48779},[3224],[507,48781,48783],{"className":48782},[2587],[507,48784,48786],{"className":48785,"style":3232},[2591],[507,48787],{},[507,48789],{"className":48790,"style":2919},[2714],[507,48792,573],{"className":48793},[2923],[507,48795],{"className":48796,"style":2919},[2714],[507,48798,48800,48804,48869,48872,48875],{"className":48799},[2561],[507,48801],{"className":48802,"style":48803},[2565],"height:2.0074em;vertical-align:-0.686em;",[507,48805,48807,48810,48866],{"className":48806},[2570],[507,48808],{"className":48809},[2941,9793],[507,48811,48813],{"className":48812},[9491],[507,48814,48816,48858],{"className":48815},[2583,3200],[507,48817,48819,48855],{"className":48818},[2587],[507,48820,48822,48833,48841],{"className":48821,"style":9806},[2591],[507,48823,48824,48827],{"style":11717},[507,48825],{"className":48826,"style":4310},[2599],[507,48828,48830],{"className":48829},[2570],[507,48831,42662],{"className":48832},[2570],[507,48834,48835,48838],{"style":9878},[507,48836],{"className":48837,"style":4310},[2599],[507,48839],{"className":48840,"style":9886},[9885],[507,48842,48843,48846],{"style":9889},[507,48844],{"className":48845,"style":4310},[2599],[507,48847,48849,48852],{"className":48848},[2570],[507,48850,584],{"className":48851},[2570],[507,48853,8563],{"className":48854,"style":2776},[2570,2611],[507,48856,3225],{"className":48857},[3224],[507,48859,48861],{"className":48860},[2587],[507,48862,48864],{"className":48863,"style":11755},[2591],[507,48865],{},[507,48867],{"className":48868},[2780,9793],[507,48870],{"className":48871,"style":2919},[2714],[507,48873,573],{"className":48874},[2923],[507,48876],{"className":48877,"style":2919},[2714],[507,48879,48881,48884,48949],{"className":48880},[2561],[507,48882],{"className":48883,"style":35838},[2565],[507,48885,48887,48890,48946],{"className":48886},[2570],[507,48888],{"className":48889},[2941,9793],[507,48891,48893],{"className":48892},[9491],[507,48894,48896,48938],{"className":48895},[2583,3200],[507,48897,48899,48935],{"className":48898},[2587],[507,48900,48902,48916,48924],{"className":48901,"style":9806},[2591],[507,48903,48904,48907],{"style":11717},[507,48905],{"className":48906,"style":4310},[2599],[507,48908,48910,48913],{"className":48909},[2570],[507,48911,42720],{"className":48912},[2570],[507,48914,8563],{"className":48915,"style":2776},[2570,2611],[507,48917,48918,48921],{"style":9878},[507,48919],{"className":48920,"style":4310},[2599],[507,48922],{"className":48923,"style":9886},[9885],[507,48925,48926,48929],{"style":9889},[507,48927],{"className":48928,"style":4310},[2599],[507,48930,48932],{"className":48931},[2570],[507,48933,625],{"className":48934},[2570],[507,48936,3225],{"className":48937},[3224],[507,48939,48941],{"className":48940},[2587],[507,48942,48944],{"className":48943,"style":29556},[2591],[507,48945],{},[507,48947],{"className":48948},[2780,9793],[507,48950,53],{"className":48951},[2570],[18,48953,48954],{},"For the same default drive strength, the full temporal period is",[507,48956,48958],{"className":48957},[2784],[507,48959,48961,48994],{"className":48960},[2523],[507,48962,48964],{"className":48963},[2527],[2529,48965,48966],{"xmlns":2531,"display":2793},[2533,48967,48968,48991],{},[2536,48969,48970,48976,48978,48981,48983,48989],{},[3168,48971,48972,48974],{},[2542,48973,37046],{},[2542,48975,20370],{},[2689,48977,573],{},[2693,48979,48980],{},"50",[6167,48982,6169],{},[2536,48984,48985,48987],{},[2542,48986,4420],{"mathvariant":2748},[2542,48988,14533],{"mathvariant":2748},[2542,48990,53],{"mathvariant":2748},[2549,48992,48993],{"encoding":2551},"T_R=50~\\mathrm{ns}.",[507,48995,48997,49052],{"className":48996,"ariaHidden":2557},[2556],[507,48998,49000,49003,49043,49046,49049],{"className":48999},[2561],[507,49001],{"className":49002,"style":3187},[2565],[507,49004,49006,49009],{"className":49005},[2570],[507,49007,37046],{"className":49008,"style":3220},[2570,2611],[507,49010,49012],{"className":49011},[2579],[507,49013,49015,49035],{"className":49014},[2583,3200],[507,49016,49018,49032],{"className":49017},[2587],[507,49019,49021],{"className":49020,"style":3207},[2591],[507,49022,49023,49026],{"style":7637},[507,49024],{"className":49025,"style":2600},[2599],[507,49027,49029],{"className":49028},[2604,2605,2606,2607],[507,49030,20370],{"className":49031,"style":20395},[2570,2611,2607],[507,49033,3225],{"className":49034},[3224],[507,49036,49038],{"className":49037},[2587],[507,49039,49041],{"className":49040,"style":3232},[2591],[507,49042],{},[507,49044],{"className":49045,"style":2919},[2714],[507,49047,573],{"className":49048},[2923],[507,49050],{"className":49051,"style":2919},[2714],[507,49053,49055,49058,49061,49064,49070],{"className":49054},[2561],[507,49056],{"className":49057,"style":2729},[2565],[507,49059,48980],{"className":49060},[2570],[507,49062,6169],{"className":49063},[2714,43889],[507,49065,49067],{"className":49066},[2570],[507,49068,48678],{"className":49069},[2570,43896],[507,49071,53],{"className":49072},[2570],[13,49074,49076],{"id":49075},"phenomenological-dephasing","Phenomenological Dephasing",[18,49078,49079],{},"The computational model includes an optional phenomenological damping factor,",[507,49081,49083],{"className":49082},[2784],[507,49084,49086,49158],{"className":49085},[2523],[507,49087,49089],{"className":49088},[2527],[2529,49090,49091],{"xmlns":2531,"display":2793},[2533,49092,49093,49155],{},[2536,49094,49095,49101,49103,49105,49107,49109,49111,49113,49119,49121,49123,49125,49127,49129,49131,49133,49153],{},[3168,49096,49097,49099],{},[2542,49098,3174],{},[2542,49100,3286],{},[2689,49102,580],{"stretchy":2755},[2542,49104,22278],{},[2689,49106,2819],{"separator":2557},[2542,49108,41704],{},[2689,49110,3649],{"stretchy":2755},[2689,49112,14934],{},[3168,49114,49115,49117],{},[2542,49116,3174],{},[2542,49118,3286],{},[2689,49120,580],{"stretchy":2755},[2542,49122,22278],{},[2689,49124,2819],{"separator":2557},[2542,49126,41704],{},[2689,49128,3649],{"stretchy":2755},[2542,49130,24069],{},[2689,49132,36690],{},[2536,49134,49135,49137,49139,49151],{},[2689,49136,580],{"fence":2557},[2689,49138,2691],{},[9491,49140,49141,49143],{},[2542,49142,41704],{},[3775,49144,49145,49147,49149],{},[2542,49146,37046],{},[2693,49148,584],{},[2689,49150,20769],{},[2689,49152,3649],{"fence":2557},[2542,49154,53],{"mathvariant":2748},[2549,49156,49157],{"encoding":2551},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[507,49159,49161,49234],{"className":49160,"ariaHidden":2557},[2556],[507,49162,49164,49167,49207,49210,49213,49216,49219,49222,49225,49228,49231],{"className":49163},[2561],[507,49165],{"className":49166,"style":2769},[2565],[507,49168,49170,49173],{"className":49169},[2570],[507,49171,3174],{"className":49172,"style":3220},[2570,2611],[507,49174,49176],{"className":49175},[2579],[507,49177,49179,49199],{"className":49178},[2583,3200],[507,49180,49182,49196],{"className":49181},[2587],[507,49183,49185],{"className":49184,"style":4507},[2591],[507,49186,49187,49190],{"style":7637},[507,49188],{"className":49189,"style":2600},[2599],[507,49191,49193],{"className":49192},[2604,2605,2606,2607],[507,49194,3286],{"className":49195},[2570,2611,2607],[507,49197,3225],{"className":49198},[3224],[507,49200,49202],{"className":49201},[2587],[507,49203,49205],{"className":49204,"style":3232},[2591],[507,49206],{},[507,49208,580],{"className":49209},[2941],[507,49211,22278],{"className":49212,"style":27338},[2570,2611],[507,49214,2819],{"className":49215},[2961],[507,49217],{"className":49218,"style":2965},[2714],[507,49220,41704],{"className":49221,"style":41776},[2570,2611],[507,49223,3649],{"className":49224},[2780],[507,49226],{"className":49227,"style":2919},[2714],[507,49229,14934],{"className":49230},[2923],[507,49232],{"className":49233,"style":2919},[2714],[507,49235,49237,49241,49281,49284,49287,49290,49293,49296,49299,49302,49305,49308,49440,49443],{"className":49236},[2561],[507,49238],{"className":49239,"style":49240},[2565],"height:2.4023em;vertical-align:-0.9523em;",[507,49242,49244,49247],{"className":49243},[2570],[507,49245,3174],{"className":49246,"style":3220},[2570,2611],[507,49248,49250],{"className":49249},[2579],[507,49251,49253,49273],{"className":49252},[2583,3200],[507,49254,49256,49270],{"className":49255},[2587],[507,49257,49259],{"className":49258,"style":4507},[2591],[507,49260,49261,49264],{"style":7637},[507,49262],{"className":49263,"style":2600},[2599],[507,49265,49267],{"className":49266},[2604,2605,2606,2607],[507,49268,3286],{"className":49269},[2570,2611,2607],[507,49271,3225],{"className":49272},[3224],[507,49274,49276],{"className":49275},[2587],[507,49277,49279],{"className":49278,"style":3232},[2591],[507,49280],{},[507,49282,580],{"className":49283},[2941],[507,49285,22278],{"className":49286,"style":27338},[2570,2611],[507,49288,2819],{"className":49289},[2961],[507,49291],{"className":49292,"style":2965},[2714],[507,49294,41704],{"className":49295,"style":41776},[2570,2611],[507,49297,3649],{"className":49298},[2780],[507,49300],{"className":49301,"style":2965},[2714],[507,49303,24069],{"className":49304},[7570],[507,49306],{"className":49307,"style":2965},[2714],[507,49309,49311,49317,49320,49434],{"className":49310},[2937],[507,49312,49314],{"className":49313,"style":2943},[2941,2942],[507,49315,580],{"className":49316},[2947,2606],[507,49318,2691],{"className":49319},[2570],[507,49321,49323,49326,49431],{"className":49322},[2570],[507,49324],{"className":49325},[2941,9793],[507,49327,49329],{"className":49328},[9491],[507,49330,49332,49422],{"className":49331},[2583,3200],[507,49333,49335,49419],{"className":49334},[2587],[507,49336,49338,49400,49408],{"className":49337,"style":48238},[2591],[507,49339,49340,49343],{"style":11717},[507,49341],{"className":49342,"style":4310},[2599],[507,49344,49346],{"className":49345},[2570],[507,49347,49349,49352],{"className":49348},[2570],[507,49350,37046],{"className":49351,"style":3220},[2570,2611],[507,49353,49355],{"className":49354},[2579],[507,49356,49358,49391],{"className":49357},[2583,3200],[507,49359,49361,49388],{"className":49360},[2587],[507,49362,49365,49377],{"className":49363,"style":49364},[2591],"height:0.6705em;",[507,49366,49368,49371],{"style":49367},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[507,49369],{"className":49370,"style":2600},[2599],[507,49372,49374],{"className":49373},[2604,2605,2606,2607],[507,49375,584],{"className":49376},[2570,2607],[507,49378,49379,49382],{"style":46894},[507,49380],{"className":49381,"style":2600},[2599],[507,49383,49385],{"className":49384},[2604,2605,2606,2607],[507,49386,20769],{"className":49387},[2719,2607],[507,49389,3225],{"className":49390},[3224],[507,49392,49394],{"className":49393},[2587],[507,49395,49398],{"className":49396,"style":49397},[2591],"height:0.2663em;",[507,49399],{},[507,49401,49402,49405],{"style":9878},[507,49403],{"className":49404,"style":4310},[2599],[507,49406],{"className":49407,"style":9886},[9885],[507,49409,49410,49413],{"style":9889},[507,49411],{"className":49412,"style":4310},[2599],[507,49414,49416],{"className":49415},[2570],[507,49417,41704],{"className":49418,"style":41776},[2570,2611],[507,49420,3225],{"className":49421},[3224],[507,49423,49425],{"className":49424},[2587],[507,49426,49429],{"className":49427,"style":49428},[2591],"height:0.9523em;",[507,49430],{},[507,49432],{"className":49433},[2780,9793],[507,49435,49437],{"className":49436,"style":2943},[2780,2942],[507,49438,3649],{"className":49439},[2947,2606],[507,49441],{"className":49442,"style":2965},[2714],[507,49444,53],{"className":49445},[2570],[18,49447,49448],{},"This exponential envelope attenuates the plotted Rabi fringes as pulse duration increases. The factor is a compact visualization device for contrast decay. A rigorous open-quantum-system treatment would require a density matrix model with relaxation channels, dephasing channels, finite temperature effects, shaped pulse envelopes, and stochastic dynamics. The default dephasing parameter is",[507,49450,49452],{"className":49451},[2784],[507,49453,49455,49490],{"className":49454},[2523],[507,49456,49458],{"className":49457},[2527],[2529,49459,49460],{"xmlns":2531,"display":2793},[2533,49461,49462,49487],{},[2536,49463,49464,49472,49474,49477,49479,49485],{},[3775,49465,49466,49468,49470],{},[2542,49467,37046],{},[2693,49469,584],{},[2689,49471,20769],{},[2689,49473,573],{},[2693,49475,49476],{},"500",[6167,49478,6169],{},[2536,49480,49481,49483],{},[2542,49482,4420],{"mathvariant":2748},[2542,49484,14533],{"mathvariant":2748},[2542,49486,53],{"mathvariant":2748},[2549,49488,49489],{"encoding":2551},"T_2^\\ast=500~\\mathrm{ns}.",[507,49491,49493,49561],{"className":49492,"ariaHidden":2557},[2556],[507,49494,49496,49500,49552,49555,49558],{"className":49495},[2561],[507,49497],{"className":49498,"style":49499},[2565],"height:0.9857em;vertical-align:-0.247em;",[507,49501,49503,49506],{"className":49502},[2570],[507,49504,37046],{"className":49505,"style":3220},[2570,2611],[507,49507,49509],{"className":49508},[2579],[507,49510,49512,49544],{"className":49511},[2583,3200],[507,49513,49515,49541],{"className":49514},[2587],[507,49516,49518,49530],{"className":49517,"style":37145},[2591],[507,49519,49521,49524],{"style":49520},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[507,49522],{"className":49523,"style":2600},[2599],[507,49525,49527],{"className":49526},[2604,2605,2606,2607],[507,49528,584],{"className":49529},[2570,2607],[507,49531,49532,49535],{"style":2906},[507,49533],{"className":49534,"style":2600},[2599],[507,49536,49538],{"className":49537},[2604,2605,2606,2607],[507,49539,20769],{"className":49540},[2719,2607],[507,49542,3225],{"className":49543},[3224],[507,49545,49547],{"className":49546},[2587],[507,49548,49550],{"className":49549,"style":3913},[2591],[507,49551],{},[507,49553],{"className":49554,"style":2919},[2714],[507,49556,573],{"className":49557},[2923],[507,49559],{"className":49560,"style":2919},[2714],[507,49562,49564,49567,49570,49573,49579],{"className":49563},[2561],[507,49565],{"className":49566,"style":2729},[2565],[507,49568,49476],{"className":49569},[2570],[507,49571,6169],{"className":49572},[2714,43889],[507,49574,49576],{"className":49575},[2570],[507,49577,48678],{"className":49578},[2570,43896],[507,49580,53],{"className":49581},[2570],[18,49583,49584,49585,49588],{},"Setting ",[504,49586,49587],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[13,49590,49592],{"id":49591},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[18,49594,49595],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit transition,",[507,49597,49599],{"className":49598},[2784],[507,49600,49602,49638],{"className":49601},[2523],[507,49603,49605],{"className":49604},[2527],[2529,49606,49607],{"xmlns":2531,"display":2793},[2533,49608,49609,49635],{},[2536,49610,49611,49613,49615,49617,49619,49621,49623,49625,49631,49633],{},[2542,49612,42745],{"mathvariant":2748},[2689,49614,573],{},[2693,49616,584],{},[2542,49618,8563],{},[2689,49620,580],{"stretchy":2755},[2542,49622,22278],{},[2689,49624,2691],{},[3168,49626,49627,49629],{},[2542,49628,22278],{},[2693,49630,601],{},[2689,49632,3649],{"stretchy":2755},[2542,49634,53],{"mathvariant":2748},[2549,49636,49637],{"encoding":2551},"\\Delta=2\\pi(f-f_0).",[507,49639,49641,49659,49686],{"className":49640,"ariaHidden":2557},[2556],[507,49642,49644,49647,49650,49653,49656],{"className":49643},[2561],[507,49645],{"className":49646,"style":2566},[2565],[507,49648,42745],{"className":49649},[2570],[507,49651],{"className":49652,"style":2919},[2714],[507,49654,573],{"className":49655},[2923],[507,49657],{"className":49658,"style":2919},[2714],[507,49660,49662,49665,49668,49671,49674,49677,49680,49683],{"className":49661},[2561],[507,49663],{"className":49664,"style":2769},[2565],[507,49666,584],{"className":49667},[2570],[507,49669,8563],{"className":49670,"style":2776},[2570,2611],[507,49672,580],{"className":49673},[2941],[507,49675,22278],{"className":49676,"style":27338},[2570,2611],[507,49678],{"className":49679,"style":2715},[2714],[507,49681,2691],{"className":49682},[2719],[507,49684],{"className":49685,"style":2715},[2714],[507,49687,49689,49692,49732,49735],{"className":49688},[2561],[507,49690],{"className":49691,"style":2769},[2565],[507,49693,49695,49698],{"className":49694},[2570],[507,49696,22278],{"className":49697,"style":27338},[2570,2611],[507,49699,49701],{"className":49700},[2579],[507,49702,49704,49724],{"className":49703},[2583,3200],[507,49705,49707,49721],{"className":49706},[2587],[507,49708,49710],{"className":49709,"style":14281},[2591],[507,49711,49712,49715],{"style":42267},[507,49713],{"className":49714,"style":2600},[2599],[507,49716,49718],{"className":49717},[2604,2605,2606,2607],[507,49719,601],{"className":49720},[2570,2607],[507,49722,3225],{"className":49723},[3224],[507,49725,49727],{"className":49726},[2587],[507,49728,49730],{"className":49729,"style":3232},[2591],[507,49731],{},[507,49733,3649],{"className":49734},[2780],[507,49736,53],{"className":49737},[2570],[18,49739,49740],{},"Expressed 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At resonance, the generalized frequency equals the applied drive strength,",[507,50287,50289],{"className":50288},[2784],[507,50290,50292,50334],{"className":50291},[2523],[507,50293,50295],{"className":50294},[2527],[2529,50296,50297],{"xmlns":2531,"display":2793},[2533,50298,50299,50331],{},[2536,50300,50301,50307,50309,50315,50317,50319,50329],{},[3168,50302,50303,50305],{},[2542,50304,42975],{},[2542,50306,20370],{},[2689,50308,580],{"stretchy":2755},[3168,50310,50311,50313],{},[2542,50312,22278],{},[2693,50314,601],{},[2689,50316,3649],{"stretchy":2755},[2689,50318,573],{},[9491,50320,50321,50323],{},[2542,50322,42662],{"mathvariant":2748},[2536,50324,50325,50327],{},[2693,50326,584],{},[2542,50328,8563],{},[2542,50330,53],{"mathvariant":2748},[2549,50332,50333],{"encoding":2551},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[507,50335,50337,50438],{"className":50336,"ariaHidden":2557},[2556],[507,50338,50340,50343,50383,50386,50426,50429,50432,50435],{"className":50339},[2561],[507,50341],{"className":50342,"style":2769},[2565],[507,50344,50346,50349],{"className":50345},[2570],[507,50347,42975],{"className":50348,"style":42996},[2570,2611],[507,50350,50352],{"className":50351},[2579],[507,50353,50355,50375],{"className":50354},[2583,3200],[507,50356,50358,50372],{"className":50357},[2587],[507,50359,50361],{"className":50360,"style":3207},[2591],[507,50362,50363,50366],{"style":43011},[507,50364],{"className":50365,"style":2600},[2599],[507,50367,50369],{"className":50368},[2604,2605,2606,2607],[507,50370,20370],{"className":50371,"style":20395},[2570,2611,2607],[507,50373,3225],{"className":50374},[3224],[507,50376,50378],{"className":50377},[2587],[507,50379,50381],{"className":50380,"style":3232},[2591],[507,50382],{},[507,50384,580],{"className":50385},[2941],[507,50387,50389,50392],{"className":50388},[2570],[507,50390,22278],{"className":50391,"style":27338},[2570,2611],[507,50393,50395],{"className":50394},[2579],[507,50396,50398,50418],{"className":50397},[2583,3200],[507,50399,50401,50415],{"className":50400},[2587],[507,50402,50404],{"className":50403,"style":14281},[2591],[507,50405,50406,50409],{"style":42267},[507,50407],{"className":50408,"style":2600},[2599],[507,50410,50412],{"className":50411},[2604,2605,2606,2607],[507,50413,601],{"className":50414},[2570,2607],[507,50416,3225],{"className":50417},[3224],[507,50419,50421],{"className":50420},[2587],[507,50422,50424],{"className":50423,"style":3232},[2591],[507,50425],{},[507,50427,3649],{"className":50428},[2780],[507,50430],{"className":50431,"style":2919},[2714],[507,50433,573],{"className":50434},[2923],[507,50436],{"className":50437,"style":2919},[2714],[507,50439,50441,50444,50509],{"className":50440},[2561],[507,50442],{"className":50443,"style":48433},[2565],[507,50445,50447,50450,50506],{"className":50446},[2570],[507,50448],{"className":50449},[2941,9793],[507,50451,50453],{"className":50452},[9491],[507,50454,50456,50498],{"className":50455},[2583,3200],[507,50457,50459,50495],{"className":50458},[2587],[507,50460,50462,50476,50484],{"className":50461,"style":47040},[2591],[507,50463,50464,50467],{"style":11717},[507,50465],{"className":50466,"style":4310},[2599],[507,50468,50470,50473],{"className":50469},[2570],[507,50471,584],{"className":50472},[2570],[507,50474,8563],{"className":50475,"style":2776},[2570,2611],[507,50477,50478,50481],{"style":9878},[507,50479],{"className":50480,"style":4310},[2599],[507,50482],{"className":50483,"style":9886},[9885],[507,50485,50486,50489],{"style":9889},[507,50487],{"className":50488,"style":4310},[2599],[507,50490,50492],{"className":50491},[2570],[507,50493,42662],{"className":50494},[2570],[507,50496,3225],{"className":50497},[3224],[507,50499,50501],{"className":50500},[2587],[507,50502,50504],{"className":50503,"style":11755},[2591],[507,50505],{},[507,50507],{"className":50508},[2780,9793],[507,50510,53],{"className":50511},[2570],[18,50513,50514,50515,50584,50585,50687],{},"As the system moves away from resonance, ",[507,50516,50518,50535],{"className":50517},[2523],[507,50519,50521],{"className":50520},[2527],[2529,50522,50523],{"xmlns":2531},[2533,50524,50525,50533],{},[2536,50526,50527],{},[3168,50528,50529,50531],{},[2542,50530,42975],{},[2542,50532,20370],{},[2549,50534,42980],{"encoding":2551},[507,50536,50538],{"className":50537,"ariaHidden":2557},[2556],[507,50539,50541,50544],{"className":50540},[2561],[507,50542],{"className":50543,"style":25197},[2565],[507,50545,50547,50550],{"className":50546},[2570],[507,50548,42975],{"className":50549,"style":42996},[2570,2611],[507,50551,50553],{"className":50552},[2579],[507,50554,50556,50576],{"className":50555},[2583,3200],[507,50557,50559,50573],{"className":50558},[2587],[507,50560,50562],{"className":50561,"style":3207},[2591],[507,50563,50564,50567],{"style":43011},[507,50565],{"className":50566,"style":2600},[2599],[507,50568,50570],{"className":50569},[2604,2605,2606,2607],[507,50571,20370],{"className":50572,"style":20395},[2570,2611,2607],[507,50574,3225],{"className":50575},[3224],[507,50577,50579],{"className":50578},[2587],[507,50580,50582],{"className":50581,"style":3232},[2591],[507,50583],{}," increases symmetrically with ",[507,50586,50588,50614],{"className":50587},[2523],[507,50589,50591],{"className":50590},[2527],[2529,50592,50593],{"xmlns":2531},[2533,50594,50595,50611],{},[2536,50596,50597,50599,50601,50603,50609],{},[2689,50598,2749],{"stretchy":2755},[2542,50600,22278],{},[2689,50602,2691],{},[3168,50604,50605,50607],{},[2542,50606,22278],{},[2693,50608,601],{},[2689,50610,2749],{"stretchy":2755},[2549,50612,50613],{"encoding":2551},"\\lvert f-f_0\\rvert",[507,50615,50617,50638],{"className":50616,"ariaHidden":2557},[2556],[507,50618,50620,50623,50626,50629,50632,50635],{"className":50619},[2561],[507,50621],{"className":50622,"style":2769},[2565],[507,50624,2749],{"className":50625},[2941],[507,50627,22278],{"className":50628,"style":27338},[2570,2611],[507,50630],{"className":50631,"style":2715},[2714],[507,50633,2691],{"className":50634},[2719],[507,50636],{"className":50637,"style":2715},[2714],[507,50639,50641,50644,50684],{"className":50640},[2561],[507,50642],{"className":50643,"style":2769},[2565],[507,50645,50647,50650],{"className":50646},[2570],[507,50648,22278],{"className":50649,"style":27338},[2570,2611],[507,50651,50653],{"className":50652},[2579],[507,50654,50656,50676],{"className":50655},[2583,3200],[507,50657,50659,50673],{"className":50658},[2587],[507,50660,50662],{"className":50661,"style":14281},[2591],[507,50663,50664,50667],{"style":42267},[507,50665],{"className":50666,"style":2600},[2599],[507,50668,50670],{"className":50669},[2604,2605,2606,2607],[507,50671,601],{"className":50672},[2570,2607],[507,50674,3225],{"className":50675},[3224],[507,50677,50679],{"className":50678},[2587],[507,50680,50682],{"className":50681,"style":3232},[2591],[507,50683],{},[507,50685,2749],{"className":50686},[2780],". The oscillations accelerate along the pulse-duration axis, and the maximum achievable excited-state probability decreases. This tradeoff is the defining visual signature of detuned Rabi dynamics.",[13,50689,50691],{"id":50690},"quantum-information-interpretation","Quantum Information Interpretation",[18,50693,50694,50695,50765],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[507,50696,50698,50716],{"className":50697},[2523],[507,50699,50701],{"className":50700},[2527],[2529,50702,50703],{"xmlns":2531},[2533,50704,50705,50713],{},[2536,50706,50707],{},[3168,50708,50709,50711],{},[2542,50710,44707],{},[2542,50712,9139],{},[2549,50714,50715],{"encoding":2551},"\\sigma_x",[507,50717,50719],{"className":50718,"ariaHidden":2557},[2556],[507,50720,50722,50725],{"className":50721},[2561],[507,50723],{"className":50724,"style":25197},[2565],[507,50726,50728,50731],{"className":50727},[2570],[507,50729,44707],{"className":50730,"style":2776},[2570,2611],[507,50732,50734],{"className":50733},[2579],[507,50735,50737,50757],{"className":50736},[2583,3200],[507,50738,50740,50754],{"className":50739},[2587],[507,50741,50743],{"className":50742,"style":4507},[2591],[507,50744,50745,50748],{"style":4572},[507,50746],{"className":50747,"style":2600},[2599],[507,50749,50751],{"className":50750},[2604,2605,2606,2607],[507,50752,9139],{"className":50753},[2570,2611,2607],[507,50755,3225],{"className":50756},[3224],[507,50758,50760],{"className":50759},[2587],[507,50761,50763],{"className":50762,"style":3232},[2591],[507,50764],{},", so the microwave pulse implements",[507,50767,50769],{"className":50768},[2784],[507,50770,50772,50829],{"className":50771},[2523],[507,50773,50775],{"className":50774},[2527],[2529,50776,50777],{"xmlns":2531,"display":2793},[2533,50778,50779,50826],{},[2536,50780,50781,50787,50789,50792,50794,50796,50798,50800,50824],{},[3168,50782,50783,50785],{},[2542,50784,20370],{},[2542,50786,9139],{},[2689,50788,580],{"stretchy":2755},[2542,50790,50791],{},"θ",[2689,50793,3649],{"stretchy":2755},[2689,50795,573],{},[2542,50797,24069],{},[2689,50799,36690],{},[2536,50801,50802,50804,50806,50822],{},[2689,50803,580],{"fence":2557},[2689,50805,2691],{},[9491,50807,50808,50820],{},[2536,50809,50810,50812,50814],{},[2542,50811,3293],{},[2542,50813,50791],{},[3168,50815,50816,50818],{},[2542,50817,44707],{},[2542,50819,9139],{},[2693,50821,584],{},[2689,50823,3649],{"fence":2557},[2689,50825,2819],{"separator":2557},[2549,50827,50828],{"encoding":2551},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[507,50830,50832,50896],{"className":50831,"ariaHidden":2557},[2556],[507,50833,50835,50838,50878,50881,50884,50887,50890,50893],{"className":50834},[2561],[507,50836],{"className":50837,"style":2769},[2565],[507,50839,50841,50844],{"className":50840},[2570],[507,50842,20370],{"className":50843,"style":20395},[2570,2611],[507,50845,50847],{"className":50846},[2579],[507,50848,50850,50870],{"className":50849},[2583,3200],[507,50851,50853,50867],{"className":50852},[2587],[507,50854,50856],{"className":50855,"style":4507},[2591],[507,50857,50858,50861],{"style":20410},[507,50859],{"className":50860,"style":2600},[2599],[507,50862,50864],{"className":50863},[2604,2605,2606,2607],[507,50865,9139],{"className":50866},[2570,2611,2607],[507,50868,3225],{"className":50869},[3224],[507,50871,50873],{"className":50872},[2587],[507,50874,50876],{"className":50875,"style":3232},[2591],[507,50877],{},[507,50879,580],{"className":50880},[2941],[507,50882,50791],{"className":50883,"style":2612},[2570,2611],[507,50885,3649],{"className":50886},[2780],[507,50888],{"className":50889,"style":2919},[2714],[507,50891,573],{"className":50892},[2923],[507,50894],{"className":50895,"style":2919},[2714],[507,50897,50899,50902,50905,50908,51031,51034],{"className":50898},[2561],[507,50900],{"className":50901,"style":37668},[2565],[507,50903,24069],{"className":50904},[7570],[507,50906],{"className":50907,"style":2965},[2714],[507,50909,50911,50917,50920,51025],{"className":50910},[2937],[507,50912,50914],{"className":50913,"style":2943},[2941,2942],[507,50915,580],{"className":50916},[2947,2606],[507,50918,2691],{"className":50919},[2570],[507,50921,50923,50926,51022],{"className":50922},[2570],[507,50924],{"className":50925},[2941,9793],[507,50927,50929],{"className":50928},[9491],[507,50930,50932,51014],{"className":50931},[2583,3200],[507,50933,50935,51011],{"className":50934},[2587],[507,50936,50938,50949,50957],{"className":50937,"style":37758},[2591],[507,50939,50940,50943],{"style":11717},[507,50941],{"className":50942,"style":4310},[2599],[507,50944,50946],{"className":50945},[2570],[507,50947,584],{"className":50948},[2570],[507,50950,50951,50954],{"style":9878},[507,50952],{"className":50953,"style":4310},[2599],[507,50955],{"className":50956,"style":9886},[9885],[507,50958,50959,50962],{"style":9889},[507,50960],{"className":50961,"style":4310},[2599],[507,50963,50965,50968,50971],{"className":50964},[2570],[507,50966,3293],{"className":50967},[2570,2611],[507,50969,50791],{"className":50970,"style":2612},[2570,2611],[507,50972,50974,50977],{"className":50973},[2570],[507,50975,44707],{"className":50976,"style":2776},[2570,2611],[507,50978,50980],{"className":50979},[2579],[507,50981,50983,51003],{"className":50982},[2583,3200],[507,50984,50986,51000],{"className":50985},[2587],[507,50987,50989],{"className":50988,"style":4507},[2591],[507,50990,50991,50994],{"style":4572},[507,50992],{"className":50993,"style":2600},[2599],[507,50995,50997],{"className":50996},[2604,2605,2606,2607],[507,50998,9139],{"className":50999},[2570,2611,2607],[507,51001,3225],{"className":51002},[3224],[507,51004,51006],{"className":51005},[2587],[507,51007,51009],{"className":51008,"style":3232},[2591],[507,51010],{},[507,51012,3225],{"className":51013},[3224],[507,51015,51017],{"className":51016},[2587],[507,51018,51020],{"className":51019,"style":11755},[2591],[507,51021],{},[507,51023],{"className":51024},[2780,9793],[507,51026,51028],{"className":51027,"style":2943},[2780,2942],[507,51029,3649],{"className":51030},[2947,2606],[507,51032],{"className":51033,"style":2965},[2714],[507,51035,2819],{"className":51036},[2961],[18,51038,47139],{},[507,51040,51042],{"className":51041},[2784],[507,51043,51045,51067],{"className":51044},[2523],[507,51046,51048],{"className":51047},[2527],[2529,51049,51050],{"xmlns":2531,"display":2793},[2533,51051,51052,51064],{},[2536,51053,51054,51056,51058,51060,51062],{},[2542,51055,50791],{},[2689,51057,573],{},[2542,51059,42662],{"mathvariant":2748},[2542,51061,41704],{},[2542,51063,53],{"mathvariant":2748},[2549,51065,51066],{"encoding":2551},"\\theta=\\Omega\\tau.",[507,51068,51070,51088],{"className":51069,"ariaHidden":2557},[2556],[507,51071,51073,51076,51079,51082,51085],{"className":51072},[2561],[507,51074],{"className":51075,"style":5434},[2565],[507,51077,50791],{"className":51078,"style":2612},[2570,2611],[507,51080],{"className":51081,"style":2919},[2714],[507,51083,573],{"className":51084},[2923],[507,51086],{"className":51087,"style":2919},[2714],[507,51089,51091,51094,51097,51100],{"className":51090},[2561],[507,51092],{"className":51093,"style":2566},[2565],[507,51095,42662],{"className":51096},[2570],[507,51098,41704],{"className":51099,"style":41776},[2570,2611],[507,51101,53],{"className":51102},[2570],[18,51104,51105,51106,51134],{},"A calibrated resonant ",[507,51107,51109,51122],{"className":51108},[2523],[507,51110,51112],{"className":51111},[2527],[2529,51113,51114],{"xmlns":2531},[2533,51115,51116,51120],{},[2536,51117,51118],{},[2542,51119,8563],{},[2549,51121,48084],{"encoding":2551},[507,51123,51125],{"className":51124,"ariaHidden":2557},[2556],[507,51126,51128,51131],{"className":51127},[2561],[507,51129],{"className":51130,"style":2639},[2565],[507,51132,8563],{"className":51133,"style":2776},[2570,2611],"-pulse enacts",[507,51136,51138],{"className":51137},[2784],[507,51139,51141,51179],{"className":51140},[2523],[507,51142,51144],{"className":51143},[2527],[2529,51145,51146],{"xmlns":2531,"display":2793},[2533,51147,51148,51176],{},[2536,51149,51150,51156,51158,51160,51162,51164,51166,51168,51174],{},[3168,51151,51152,51154],{},[2542,51153,20370],{},[2542,51155,9139],{},[2689,51157,580],{"stretchy":2755},[2542,51159,8563],{},[2689,51161,3649],{"stretchy":2755},[2689,51163,573],{},[2689,51165,2691],{},[2542,51167,3293],{},[3168,51169,51170,51172],{},[2542,51171,44707],{},[2542,51173,9139],{},[2689,51175,2819],{"separator":2557},[2549,51177,51178],{"encoding":2551},"R_x(\\pi)=-i\\sigma_x,",[507,51180,51182,51246],{"className":51181,"ariaHidden":2557},[2556],[507,51183,51185,51188,51228,51231,51234,51237,51240,51243],{"className":51184},[2561],[507,51186],{"className":51187,"style":2769},[2565],[507,51189,51191,51194],{"className":51190},[2570],[507,51192,20370],{"className":51193,"style":20395},[2570,2611],[507,51195,51197],{"className":51196},[2579],[507,51198,51200,51220],{"className":51199},[2583,3200],[507,51201,51203,51217],{"className":51202},[2587],[507,51204,51206],{"className":51205,"style":4507},[2591],[507,51207,51208,51211],{"style":20410},[507,51209],{"className":51210,"style":2600},[2599],[507,51212,51214],{"className":51213},[2604,2605,2606,2607],[507,51215,9139],{"className":51216},[2570,2611,2607],[507,51218,3225],{"className":51219},[3224],[507,51221,51223],{"className":51222},[2587],[507,51224,51226],{"className":51225,"style":3232},[2591],[507,51227],{},[507,51229,580],{"className":51230},[2941],[507,51232,8563],{"className":51233,"style":2776},[2570,2611],[507,51235,3649],{"className":51236},[2780],[507,51238],{"className":51239,"style":2919},[2714],[507,51241,573],{"className":51242},[2923],[507,51244],{"className":51245,"style":2919},[2714],[507,51247,51249,51252,51255,51258,51298],{"className":51248},[2561],[507,51250],{"className":51251,"style":27986},[2565],[507,51253,2691],{"className":51254},[2570],[507,51256,3293],{"className":51257},[2570,2611],[507,51259,51261,51264],{"className":51260},[2570],[507,51262,44707],{"className":51263,"style":2776},[2570,2611],[507,51265,51267],{"className":51266},[2579],[507,51268,51270,51290],{"className":51269},[2583,3200],[507,51271,51273,51287],{"className":51272},[2587],[507,51274,51276],{"className":51275,"style":4507},[2591],[507,51277,51278,51281],{"style":4572},[507,51279],{"className":51280,"style":2600},[2599],[507,51282,51284],{"className":51283},[2604,2605,2606,2607],[507,51285,9139],{"className":51286},[2570,2611,2607],[507,51288,3225],{"className":51289},[3224],[507,51291,51293],{"className":51292},[2587],[507,51294,51296],{"className":51295,"style":3232},[2591],[507,51297],{},[507,51299,2819],{"className":51300},[2961],[18,51302,51303,51304,51332,51333,51369],{},"which corresponds to a quantum ",[507,51305,51307,51320],{"className":51306},[2523],[507,51308,51310],{"className":51309},[2527],[2529,51311,51312],{"xmlns":2531},[2533,51313,51314,51318],{},[2536,51315,51316],{},[2542,51317,7731],{},[2549,51319,7731],{"encoding":2551},[507,51321,51323],{"className":51322,"ariaHidden":2557},[2556],[507,51324,51326,51329],{"className":51325},[2561],[507,51327],{"className":51328,"style":2566},[2565],[507,51330,7731],{"className":51331,"style":7876},[2570,2611]," gate up to a global phase. A resonant ",[507,51334,51336,51354],{"className":51335},[2523],[507,51337,51339],{"className":51338},[2527],[2529,51340,51341],{"xmlns":2531},[2533,51342,51343,51351],{},[2536,51344,51345,51347,51349],{},[2542,51346,8563],{},[2542,51348,645],{"mathvariant":2748},[2693,51350,584],{},[2549,51352,51353],{"encoding":2551},"\\pi\u002F2",[507,51355,51357],{"className":51356,"ariaHidden":2557},[2556],[507,51358,51360,51363,51366],{"className":51359},[2561],[507,51361],{"className":51362,"style":2769},[2565],[507,51364,8563],{"className":51365,"style":2776},[2570,2611],[507,51367,26121],{"className":51368},[2570],"-pulse prepares a balanced coherent superposition,",[507,51371,51373],{"className":51372},[2784],[507,51374,51376,51422],{"className":51375},[2523],[507,51377,51379],{"className":51378},[2527],[2529,51380,51381],{"xmlns":2531,"display":2793},[2533,51382,51383,51419],{},[2536,51384,51385,51387,51389,51391,51393,51417],{},[2689,51386,2749],{"stretchy":2755},[2542,51388,37913],{},[2689,51390,2756],{"stretchy":2755},[2689,51392,14934],{},[9491,51394,51395,51413],{},[2536,51396,51397,51399,51401,51403,51405,51407,51409,51411],{},[2689,51398,2749],{"stretchy":2755},[2542,51400,37913],{},[2689,51402,2756],{"stretchy":2755},[2689,51404,2691],{},[2542,51406,3293],{},[2689,51408,2749],{"stretchy":2755},[2542,51410,3286],{},[2689,51412,2756],{"stretchy":2755},[9496,51414,51415],{},[2693,51416,584],{},[2689,51418,2819],{"separator":2557},[2549,51420,51421],{"encoding":2551},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[507,51423,51425,51449],{"className":51424,"ariaHidden":2557},[2556],[507,51426,51428,51431,51434,51437,51440,51443,51446],{"className":51427},[2561],[507,51429],{"className":51430,"style":2769},[2565],[507,51432,2749],{"className":51433},[2941],[507,51435,37913],{"className":51436,"style":2776},[2570,2611],[507,51438,2756],{"className":51439},[2780],[507,51441],{"className":51442,"style":2919},[2714],[507,51444,14934],{"className":51445},[2923],[507,51447],{"className":51448,"style":2919},[2714],[507,51450,51452,51456,51589],{"className":51451},[2561],[507,51453],{"className":51454,"style":51455},[2565],"height:2.357em;vertical-align:-0.93em;",[507,51457,51459,51462,51586],{"className":51458},[2570],[507,51460],{"className":51461},[2941,9793],[507,51463,51465],{"className":51464},[9491],[507,51466,51468,51578],{"className":51467},[2583,3200],[507,51469,51471,51575],{"className":51470},[2587],[507,51472,51474,51529,51537],{"className":51473,"style":29364},[2591],[507,51475,51476,51479],{"style":9809},[507,51477],{"className":51478,"style":4310},[2599],[507,51480,51482],{"className":51481},[2570],[507,51483,51485],{"className":51484},[2570,9819],[507,51486,51488,51521],{"className":51487},[2583,3200],[507,51489,51491,51518],{"className":51490},[2587],[507,51492,51494,51506],{"className":51493,"style":9829},[2591],[507,51495,51497,51500],{"className":51496,"style":4306},[9833],[507,51498],{"className":51499,"style":4310},[2599],[507,51501,51503],{"className":51502,"style":9840},[2570],[507,51504,584],{"className":51505},[2570],[507,51507,51508,51511],{"style":9846},[507,51509],{"className":51510,"style":4310},[2599],[507,51512,51514],{"className":51513,"style":9854},[9853],[6281,51515,51516],{"xmlns":6283,"width":9857,"height":9858,"viewBox":9859,"preserveAspectRatio":9860},[6290,51517],{"d":9863},[507,51519,3225],{"className":51520},[3224],[507,51522,51524],{"className":51523},[2587],[507,51525,51527],{"className":51526,"style":9873},[2591],[507,51528],{},[507,51530,51531,51534],{"style":9878},[507,51532],{"className":51533,"style":4310},[2599],[507,51535],{"className":51536,"style":9886},[9885],[507,51538,51539,51542],{"style":9889},[507,51540],{"className":51541,"style":4310},[2599],[507,51543,51545,51548,51551,51554,51557,51560,51563,51566,51569,51572],{"className":51544},[2570],[507,51546,2749],{"className":51547},[2941],[507,51549,37913],{"className":51550,"style":2776},[2570,2611],[507,51552,2756],{"className":51553},[2780],[507,51555],{"className":51556,"style":2715},[2714],[507,51558,2691],{"className":51559},[2719],[507,51561],{"className":51562,"style":2715},[2714],[507,51564,3293],{"className":51565},[2570,2611],[507,51567,2749],{"className":51568},[2941],[507,51570,3286],{"className":51571},[2570,2611],[507,51573,2756],{"className":51574},[2780],[507,51576,3225],{"className":51577},[3224],[507,51579,51581],{"className":51580},[2587],[507,51582,51584],{"className":51583,"style":9908},[2591],[507,51585],{},[507,51587],{"className":51588},[2780,9793],[507,51590,2819],{"className":51591},[2961],[18,51593,51594],{},"with a phase convention set by the microwave drive phase.",[18,51596,51597],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[507,51599,51601],{"className":51600},[2784],[507,51602,51604,51674],{"className":51603},[2523],[507,51605,51607],{"className":51606},[2527],[2529,51608,51609],{"xmlns":2531,"display":2793},[2533,51610,51611,51671],{},[2536,51612,51613,51624,51626,51628,51630,51632,51634,51636,51669],{},[3168,51614,51615,51617],{},[2542,51616,20370],{},[4271,51618,51619,51621],{"accent":2557},[2542,51620,4420],{},[2689,51622,51623],{},"^",[2689,51625,580],{"stretchy":2755},[2542,51627,50791],{},[2689,51629,3649],{"stretchy":2755},[2689,51631,573],{},[2542,51633,24069],{},[2689,51635,36690],{},[2536,51637,51638,51640,51642,51652,51658,51661,51667],{},[2689,51639,580],{"fence":2557},[2689,51641,2691],{},[9491,51643,51644,51650],{},[2536,51645,51646,51648],{},[2542,51647,3293],{},[2542,51649,50791],{},[2693,51651,584],{},[4271,51653,51654,51656],{"accent":2557},[2542,51655,4420],{},[2689,51657,51623],{},[2689,51659,51660],{},"⋅",[4271,51662,51663,51665],{"accent":2557},[2542,51664,44707],{},[2689,51666,6177],{},[2689,51668,3649],{"fence":2557},[2689,51670,2819],{"separator":2557},[2549,51672,51673],{"encoding":2551},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[507,51675,51677,51774],{"className":51676,"ariaHidden":2557},[2556],[507,51678,51680,51683,51756,51759,51762,51765,51768,51771],{"className":51679},[2561],[507,51681],{"className":51682,"style":2769},[2565],[507,51684,51686,51689],{"className":51685},[2570],[507,51687,20370],{"className":51688,"style":20395},[2570,2611],[507,51690,51692],{"className":51691},[2579],[507,51693,51695,51748],{"className":51694},[2583,3200],[507,51696,51698,51745],{"className":51697},[2587],[507,51699,51701],{"className":51700,"style":11243},[2591],[507,51702,51703,51706],{"style":20410},[507,51704],{"className":51705,"style":2600},[2599],[507,51707,51709],{"className":51708},[2604,2605,2606,2607],[507,51710,51712],{"className":51711},[2570,2607],[507,51713,51715],{"className":51714},[2570,4294,2607],[507,51716,51718],{"className":51717},[2583],[507,51719,51721],{"className":51720},[2587],[507,51722,51724,51733],{"className":51723,"style":5434},[2591],[507,51725,51727,51730],{"style":51726},"top:-2.7em;",[507,51728],{"className":51729,"style":2600},[2599],[507,51731,4420],{"className":51732},[2570,2611,2607],[507,51734,51735,51738],{"style":51726},[507,51736],{"className":51737,"style":2600},[2599],[507,51739,51742],{"className":51740,"style":51741},[4323],"left:-0.25em;",[507,51743,51623],{"className":51744},[2570,2607],[507,51746,3225],{"className":51747},[3224],[507,51749,51751],{"className":51750},[2587],[507,51752,51754],{"className":51753,"style":3232},[2591],[507,51755],{},[507,51757,580],{"className":51758},[2941],[507,51760,50791],{"className":51761,"style":2612},[2570,2611],[507,51763,3649],{"className":51764},[2780],[507,51766],{"className":51767,"style":2919},[2714],[507,51769,573],{"className":51770},[2923],[507,51772],{"className":51773,"style":2919},[2714],[507,51775,51777,51780,51783,51786,51945,51948],{"className":51776},[2561],[507,51778],{"className":51779,"style":37668},[2565],[507,51781,24069],{"className":51782},[7570],[507,51784],{"className":51785,"style":2965},[2714],[507,51787,51789,51795,51798,51863,51894,51897,51900,51903,51939],{"className":51788},[2937],[507,51790,51792],{"className":51791,"style":2943},[2941,2942],[507,51793,580],{"className":51794},[2947,2606],[507,51796,2691],{"className":51797},[2570],[507,51799,51801,51804,51860],{"className":51800},[2570],[507,51802],{"className":51803},[2941,9793],[507,51805,51807],{"className":51806},[9491],[507,51808,51810,51852],{"className":51809},[2583,3200],[507,51811,51813,51849],{"className":51812},[2587],[507,51814,51816,51827,51835],{"className":51815,"style":37758},[2591],[507,51817,51818,51821],{"style":11717},[507,51819],{"className":51820,"style":4310},[2599],[507,51822,51824],{"className":51823},[2570],[507,51825,584],{"className":51826},[2570],[507,51828,51829,51832],{"style":9878},[507,51830],{"className":51831,"style":4310},[2599],[507,51833],{"className":51834,"style":9886},[9885],[507,51836,51837,51840],{"style":9889},[507,51838],{"className":51839,"style":4310},[2599],[507,51841,51843,51846],{"className":51842},[2570],[507,51844,3293],{"className":51845},[2570,2611],[507,51847,50791],{"className":51848,"style":2612},[2570,2611],[507,51850,3225],{"className":51851},[3224],[507,51853,51855],{"className":51854},[2587],[507,51856,51858],{"className":51857,"style":11755},[2591],[507,51859],{},[507,51861],{"className":51862},[2780,9793],[507,51864,51866],{"className":51865},[2570,4294],[507,51867,51869],{"className":51868},[2583],[507,51870,51872],{"className":51871},[2587],[507,51873,51875,51883],{"className":51874,"style":5434},[2591],[507,51876,51877,51880],{"style":4306},[507,51878],{"className":51879,"style":4310},[2599],[507,51881,4420],{"className":51882},[2570,2611],[507,51884,51885,51888],{"style":4306},[507,51886],{"className":51887,"style":4310},[2599],[507,51889,51891],{"className":51890,"style":51741},[4323],[507,51892,51623],{"className":51893},[2570],[507,51895],{"className":51896,"style":2715},[2714],[507,51898,51660],{"className":51899},[2719],[507,51901],{"className":51902,"style":2715},[2714],[507,51904,51906],{"className":51905},[2570,4294],[507,51907,51909],{"className":51908},[2583],[507,51910,51912],{"className":51911},[2587],[507,51913,51915,51923],{"className":51914,"style":6257},[2591],[507,51916,51917,51920],{"style":4306},[507,51918],{"className":51919,"style":4310},[2599],[507,51921,44707],{"className":51922,"style":2776},[2570,2611],[507,51924,51925,51928],{"style":4306},[507,51926],{"className":51927,"style":4310},[2599],[507,51929,51932],{"className":51930,"style":51931},[4323],"left:-0.2355em;",[507,51933,51935],{"className":51934,"style":6279},[6278],[6281,51936,51937],{"xmlns":6283,"width":6284,"height":6285,"style":6286,"viewBox":6287,"preserveAspectRatio":6288},[6290,51938],{"d":6292},[507,51940,51942],{"className":51941,"style":2943},[2780,2942],[507,51943,3649],{"className":51944},[2947,2606],[507,51946],{"className":51947,"style":2965},[2714],[507,51949,2819],{"className":51950},[2961],[18,51952,47139],{},[507,51954,51956],{"className":51955},[2784],[507,51957,51959,52021],{"className":51958},[2523],[507,51960,51962],{"className":51961},[2527],[2529,51963,51964],{"xmlns":2531,"display":2793},[2533,51965,51966,52018],{},[2536,51967,51968,51974,51976,51986,51988,51990,51992,51994,51996,51998,52000,52002,52004,52006,52008,52014,52016],{},[4271,51969,51970,51972],{"accent":2557},[2542,51971,4420],{},[2689,51973,51623],{},[2689,51975,573],{},[9491,51977,51978,51980],{},[2693,51979,625],{},[3168,51981,51982,51984],{},[2542,51983,42662],{"mathvariant":2748},[2542,51985,20370],{},[2689,51987,580],{"stretchy":2755},[2542,51989,42662],{"mathvariant":2748},[2689,51991,2819],{"separator":2557},[2693,51993,601],{},[2689,51995,2819],{"separator":2557},[2542,51997,42745],{"mathvariant":2748},[2689,51999,3649],{"stretchy":2755},[2689,52001,2819],{"separator":2557},[2714,52003],{"width":18011},[2542,52005,50791],{},[2689,52007,573],{},[3168,52009,52010,52012],{},[2542,52011,42662],{"mathvariant":2748},[2542,52013,20370],{},[2542,52015,41704],{},[2542,52017,53],{"mathvariant":2748},[2549,52019,52020],{"encoding":2551},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[507,52022,52024,52070,52225],{"className":52023,"ariaHidden":2557},[2556],[507,52025,52027,52030,52061,52064,52067],{"className":52026},[2561],[507,52028],{"className":52029,"style":5434},[2565],[507,52031,52033],{"className":52032},[2570,4294],[507,52034,52036],{"className":52035},[2583],[507,52037,52039],{"className":52038},[2587],[507,52040,52042,52050],{"className":52041,"style":5434},[2591],[507,52043,52044,52047],{"style":4306},[507,52045],{"className":52046,"style":4310},[2599],[507,52048,4420],{"className":52049},[2570,2611],[507,52051,52052,52055],{"style":4306},[507,52053],{"className":52054,"style":4310},[2599],[507,52056,52058],{"className":52057,"style":51741},[4323],[507,52059,51623],{"className":52060},[2570],[507,52062],{"className":52063,"style":2919},[2714],[507,52065,573],{"className":52066},[2923],[507,52068],{"className":52069,"style":2919},[2714],[507,52071,52073,52077,52177,52180,52183,52186,52189,52192,52195,52198,52201,52204,52207,52210,52213,52216,52219,52222],{"className":52072},[2561],[507,52074],{"className":52075,"style":52076},[2565],"height:2.1574em;vertical-align:-0.836em;",[507,52078,52080,52083,52174],{"className":52079},[2570],[507,52081],{"className":52082},[2941,9793],[507,52084,52086],{"className":52085},[9491],[507,52087,52089,52165],{"className":52088},[2583,3200],[507,52090,52092,52162],{"className":52091},[2587],[507,52093,52095,52143,52151],{"className":52094,"style":9806},[2591],[507,52096,52097,52100],{"style":11717},[507,52098],{"className":52099,"style":4310},[2599],[507,52101,52103],{"className":52102},[2570],[507,52104,52106,52109],{"className":52105},[2570],[507,52107,42662],{"className":52108},[2570],[507,52110,52112],{"className":52111},[2579],[507,52113,52115,52135],{"className":52114},[2583,3200],[507,52116,52118,52132],{"className":52117},[2587],[507,52119,52121],{"className":52120,"style":3207},[2591],[507,52122,52123,52126],{"style":5398},[507,52124],{"className":52125,"style":2600},[2599],[507,52127,52129],{"className":52128},[2604,2605,2606,2607],[507,52130,20370],{"className":52131,"style":20395},[2570,2611,2607],[507,52133,3225],{"className":52134},[3224],[507,52136,52138],{"className":52137},[2587],[507,52139,52141],{"className":52140,"style":3232},[2591],[507,52142],{},[507,52144,52145,52148],{"style":9878},[507,52146],{"className":52147,"style":4310},[2599],[507,52149],{"className":52150,"style":9886},[9885],[507,52152,52153,52156],{"style":9889},[507,52154],{"className":52155,"style":4310},[2599],[507,52157,52159],{"className":52158},[2570],[507,52160,625],{"className":52161},[2570],[507,52163,3225],{"className":52164},[3224],[507,52166,52168],{"className":52167},[2587],[507,52169,52172],{"className":52170,"style":52171},[2591],"height:0.836em;",[507,52173],{},[507,52175],{"className":52176},[2780,9793],[507,52178,580],{"className":52179},[2941],[507,52181,42662],{"className":52182},[2570],[507,52184,2819],{"className":52185},[2961],[507,52187],{"className":52188,"style":2965},[2714],[507,52190,601],{"className":52191},[2570],[507,52193,2819],{"className":52194},[2961],[507,52196],{"className":52197,"style":2965},[2714],[507,52199,42745],{"className":52200},[2570],[507,52202,3649],{"className":52203},[2780],[507,52205,2819],{"className":52206},[2961],[507,52208],{"className":52209,"style":18629},[2714],[507,52211],{"className":52212,"style":2965},[2714],[507,52214,50791],{"className":52215,"style":2612},[2570,2611],[507,52217],{"className":52218,"style":2919},[2714],[507,52220,573],{"className":52221},[2923],[507,52223],{"className":52224,"style":2919},[2714],[507,52226,52228,52231,52271,52274],{"className":52227},[2561],[507,52229],{"className":52230,"style":3187},[2565],[507,52232,52234,52237],{"className":52233},[2570],[507,52235,42662],{"className":52236},[2570],[507,52238,52240],{"className":52239},[2579],[507,52241,52243,52263],{"className":52242},[2583,3200],[507,52244,52246,52260],{"className":52245},[2587],[507,52247,52249],{"className":52248,"style":3207},[2591],[507,52250,52251,52254],{"style":5398},[507,52252],{"className":52253,"style":2600},[2599],[507,52255,52257],{"className":52256},[2604,2605,2606,2607],[507,52258,20370],{"className":52259,"style":20395},[2570,2611,2607],[507,52261,3225],{"className":52262},[3224],[507,52264,52266],{"className":52265},[2587],[507,52267,52269],{"className":52268,"style":3232},[2591],[507,52270],{},[507,52272,41704],{"className":52273,"style":41776},[2570,2611],[507,52275,53],{"className":52276},[2570],[18,52278,52279],{},"This representation shows that detuning increases the rotation rate and tilts the control axis. Under this idealized model, the qubit still evolves coherently, although the trajectory generally misses the pure excited-state pole.",[13,52281,52283],{"id":52282},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[18,52285,52286],{},"A Bloch sphere maps any pure qubit state to a unit vector in three-dimensional space. Ground and excited states occupy opposite poles. Equal-amplitude coherent superpositions lie on the equator.",[18,52288,52289,52290,52328,52329,52357,52358,52393],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[507,52291,52293,52310],{"className":52292},[2523],[507,52294,52296],{"className":52295},[2527],[2529,52297,52298],{"xmlns":2531},[2533,52299,52300,52308],{},[2536,52301,52302,52304,52306],{},[2689,52303,2749],{"stretchy":2755},[2542,52305,37913],{},[2689,52307,2756],{"stretchy":2755},[2549,52309,41895],{"encoding":2551},[507,52311,52313],{"className":52312,"ariaHidden":2557},[2556],[507,52314,52316,52319,52322,52325],{"className":52315},[2561],[507,52317],{"className":52318,"style":2769},[2565],[507,52320,2749],{"className":52321},[2941],[507,52323,37913],{"className":52324,"style":2776},[2570,2611],[507,52326,2756],{"className":52327},[2780],", a properly timed ",[507,52330,52332,52345],{"className":52331},[2523],[507,52333,52335],{"className":52334},[2527],[2529,52336,52337],{"xmlns":2531},[2533,52338,52339,52343],{},[2536,52340,52341],{},[2542,52342,8563],{},[2549,52344,48084],{"encoding":2551},[507,52346,52348],{"className":52347,"ariaHidden":2557},[2556],[507,52349,52351,52354],{"className":52350},[2561],[507,52352],{"className":52353,"style":2639},[2565],[507,52355,8563],{"className":52356,"style":2776},[2570,2611],"-pulse moves the state to the excited-state pole. A ",[507,52359,52361,52378],{"className":52360},[2523],[507,52362,52364],{"className":52363},[2527],[2529,52365,52366],{"xmlns":2531},[2533,52367,52368,52376],{},[2536,52369,52370,52372,52374],{},[2542,52371,8563],{},[2542,52373,645],{"mathvariant":2748},[2693,52375,584],{},[2549,52377,51353],{"encoding":2551},[507,52379,52381],{"className":52380,"ariaHidden":2557},[2556],[507,52382,52384,52387,52390],{"className":52383},[2561],[507,52385],{"className":52386,"style":2769},[2565],[507,52388,8563],{"className":52389,"style":2776},[2570,2611],[507,52391,26121],{"className":52392},[2570],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[18,52395,52396,52397,52425],{},"Detuning tilts the rotation axis toward the ",[507,52398,52400,52413],{"className":52399},[2523],[507,52401,52403],{"className":52402},[2527],[2529,52404,52405],{"xmlns":2531},[2533,52406,52407,52411],{},[2536,52408,52409],{},[2542,52410,666],{},[2549,52412,666],{"encoding":2551},[507,52414,52416],{"className":52415,"ariaHidden":2557},[2556],[507,52417,52419,52422],{"className":52418},[2561],[507,52420],{"className":52421,"style":2639},[2565],[507,52423,666],{"className":52424,"style":20582},[2570,2611],"-axis. The state vector then precesses around the tilted axis and generally fails to reach the excited-state pole. This geometric picture explains the amplitude coefficient",[507,52427,52429],{"className":52428},[2784],[507,52430,52432,52462],{"className":52431},[2523],[507,52433,52435],{"className":52434},[2527],[2529,52436,52437],{"xmlns":2531,"display":2793},[2533,52438,52439,52459],{},[2536,52440,52441,52457],{},[9491,52442,52443,52449],{},[2539,52444,52445,52447],{},[2542,52446,42662],{"mathvariant":2748},[2693,52448,584],{},[3775,52450,52451,52453,52455],{},[2542,52452,42662],{"mathvariant":2748},[2542,52454,20370],{},[2693,52456,584],{},[2542,52458,53],{"mathvariant":2748},[2549,52460,52461],{"encoding":2551},"\\frac{\\Omega^2}{\\Omega_R^2}.",[507,52463,52465],{"className":52464,"ariaHidden":2557},[2556],[507,52466,52468,52471,52607],{"className":52467},[2561],[507,52469],{"className":52470,"style":46833},[2565],[507,52472,52474,52477,52604],{"className":52473},[2570],[507,52475],{"className":52476},[2941,9793],[507,52478,52480],{"className":52479},[9491],[507,52481,52483,52596],{"className":52482},[2583,3200],[507,52484,52486,52593],{"className":52485},[2587],[507,52487,52489,52548,52556],{"className":52488,"style":46852},[2591],[507,52490,52491,52494],{"style":11717},[507,52492],{"className":52493,"style":4310},[2599],[507,52495,52497],{"className":52496},[2570],[507,52498,52500,52503],{"className":52499},[2570],[507,52501,42662],{"className":52502},[2570],[507,52504,52506],{"className":52505},[2579],[507,52507,52509,52540],{"className":52508},[2583,3200],[507,52510,52512,52537],{"className":52511},[2587],[507,52513,52515,52526],{"className":52514,"style":46879},[2591],[507,52516,52517,52520],{"style":46882},[507,52518],{"className":52519,"style":2600},[2599],[507,52521,52523],{"className":52522},[2604,2605,2606,2607],[507,52524,20370],{"className":52525,"style":20395},[2570,2611,2607],[507,52527,52528,52531],{"style":46894},[507,52529],{"className":52530,"style":2600},[2599],[507,52532,52534],{"className":52533},[2604,2605,2606,2607],[507,52535,584],{"className":52536},[2570,2607],[507,52538,3225],{"className":52539},[3224],[507,52541,52543],{"className":52542},[2587],[507,52544,52546],{"className":52545,"style":46913},[2591],[507,52547],{},[507,52549,52550,52553],{"style":9878},[507,52551],{"className":52552,"style":4310},[2599],[507,52554],{"className":52555,"style":9886},[9885],[507,52557,52558,52561],{"style":9889},[507,52559],{"className":52560,"style":4310},[2599],[507,52562,52564],{"className":52563},[2570],[507,52565,52567,52570],{"className":52566},[2570],[507,52568,42662],{"className":52569},[2570],[507,52571,52573],{"className":52572},[2579],[507,52574,52576],{"className":52575},[2583],[507,52577,52579],{"className":52578},[2587],[507,52580,52582],{"className":52581,"style":13224},[2591],[507,52583,52584,52587],{"style":2595},[507,52585],{"className":52586,"style":2600},[2599],[507,52588,52590],{"className":52589},[2604,2605,2606,2607],[507,52591,584],{"className":52592},[2570,2607],[507,52594,3225],{"className":52595},[3224],[507,52597,52599],{"className":52598},[2587],[507,52600,52602],{"className":52601,"style":46970},[2591],[507,52603],{},[507,52605],{"className":52606},[2780,9793],[507,52608,53],{"className":52609},[2570],[18,52611,52612,52613,52652],{},"As ",[507,52614,52616,52634],{"className":52615},[2523],[507,52617,52619],{"className":52618},[2527],[2529,52620,52621],{"xmlns":2531},[2533,52622,52623,52631],{},[2536,52624,52625,52627,52629],{},[2689,52626,2749],{"stretchy":2755},[2542,52628,42745],{"mathvariant":2748},[2689,52630,2749],{"stretchy":2755},[2549,52632,52633],{"encoding":2551},"\\lvert\\Delta\\rvert",[507,52635,52637],{"className":52636,"ariaHidden":2557},[2556],[507,52638,52640,52643,52646,52649],{"className":52639},[2561],[507,52641],{"className":52642,"style":2769},[2565],[507,52644,2749],{"className":52645},[2941],[507,52647,42745],{"className":52648},[2570],[507,52650,2749],{"className":52651},[2780]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[13,52654,52656],{"id":52655},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[18,52658,52659,52660,52688,52689,52787],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[507,52661,52663,52676],{"className":52662},[2523],[507,52664,52666],{"className":52665},[2527],[2529,52667,52668],{"xmlns":2531},[2533,52669,52670,52674],{},[2536,52671,52672],{},[2542,52673,22278],{},[2549,52675,22278],{"encoding":2551},[507,52677,52679],{"className":52678,"ariaHidden":2557},[2556],[507,52680,52682,52685],{"className":52681},[2561],[507,52683],{"className":52684,"style":7035},[2565],[507,52686,22278],{"className":52687,"style":27338},[2570,2611],", the trace ",[507,52690,52692,52720],{"className":52691},[2523],[507,52693,52695],{"className":52694},[2527],[2529,52696,52697],{"xmlns":2531},[2533,52698,52699,52717],{},[2536,52700,52701,52707,52709,52711,52713,52715],{},[3168,52702,52703,52705],{},[2542,52704,3174],{},[2542,52706,3286],{},[2689,52708,580],{"stretchy":2755},[2542,52710,41704],{},[2689,52712,2819],{"separator":2557},[2542,52714,22278],{},[2689,52716,3649],{"stretchy":2755},[2549,52718,52719],{"encoding":2551},"P_e(\\tau,f)",[507,52721,52723],{"className":52722,"ariaHidden":2557},[2556],[507,52724,52726,52729,52769,52772,52775,52778,52781,52784],{"className":52725},[2561],[507,52727],{"className":52728,"style":2769},[2565],[507,52730,52732,52735],{"className":52731},[2570],[507,52733,3174],{"className":52734,"style":3220},[2570,2611],[507,52736,52738],{"className":52737},[2579],[507,52739,52741,52761],{"className":52740},[2583,3200],[507,52742,52744,52758],{"className":52743},[2587],[507,52745,52747],{"className":52746,"style":4507},[2591],[507,52748,52749,52752],{"style":7637},[507,52750],{"className":52751,"style":2600},[2599],[507,52753,52755],{"className":52754},[2604,2605,2606,2607],[507,52756,3286],{"className":52757},[2570,2611,2607],[507,52759,3225],{"className":52760},[3224],[507,52762,52764],{"className":52763},[2587],[507,52765,52767],{"className":52766,"style":3232},[2591],[507,52768],{},[507,52770,580],{"className":52771},[2941],[507,52773,41704],{"className":52774,"style":41776},[2570,2611],[507,52776,2819],{"className":52777},[2961],[507,52779],{"className":52780,"style":2965},[2714],[507,52782,22278],{"className":52783,"style":27338},[2570,2611],[507,52785,3649],{"className":52786},[2780]," is treated as a time-domain signal. The spectral amplitude is",[507,52789,52791],{"className":52790},[2784],[507,52792,52794,52857],{"className":52793},[2523],[507,52795,52797],{"className":52796},[2527],[2529,52798,52799],{"xmlns":2531,"display":2793},[2533,52800,52801,52854],{},[2536,52802,52803,52805,52807,52809,52811,52813,52815,52817,52852],{},[2542,52804,5857],{},[2689,52806,580],{"stretchy":2755},[2542,52808,42975],{},[2689,52810,2819],{"separator":2557},[2542,52812,22278],{},[2689,52814,3649],{"stretchy":2755},[2689,52816,573],{},[2536,52818,52819,52821,52828,52850],{},[2689,52820,2749],{"fence":2557},[3168,52822,52823,52826],{},[2542,52824,52825],{"mathvariant":2544},"F",[2542,52827,41704],{},[2536,52829,52830,52832,52838,52840,52842,52844,52846,52848],{},[2689,52831,2810],{"fence":2557},[3168,52833,52834,52836],{},[2542,52835,3174],{},[2542,52837,3286],{},[2689,52839,580],{"stretchy":2755},[2542,52841,41704],{},[2689,52843,2819],{"separator":2557},[2542,52845,22278],{},[2689,52847,3649],{"stretchy":2755},[2689,52849,2872],{"fence":2557},[2689,52851,2749],{"fence":2557},[2542,52853,53],{"mathvariant":2748},[2549,52855,52856],{"encoding":2551},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[507,52858,52860,52896],{"className":52859,"ariaHidden":2557},[2556],[507,52861,52863,52866,52869,52872,52875,52878,52881,52884,52887,52890,52893],{"className":52862},[2561],[507,52864],{"className":52865,"style":2769},[2565],[507,52867,5857],{"className":52868,"style":5892},[2570,2611],[507,52870,580],{"className":52871},[2941],[507,52873,42975],{"className":52874,"style":42996},[2570,2611],[507,52876,2819],{"className":52877},[2961],[507,52879],{"className":52880,"style":2965},[2714],[507,52882,22278],{"className":52883,"style":27338},[2570,2611],[507,52885,3649],{"className":52886},[2780],[507,52888],{"className":52889,"style":2919},[2714],[507,52891,573],{"className":52892},[2923],[507,52894],{"className":52895,"style":2919},[2714],[507,52897,52899,52902,53026,53029],{"className":52898},[2561],[507,52900],{"className":52901,"style":2769},[2565],[507,52903,52905,52908,52953,52956,53023],{"className":52904},[2937],[507,52906,2749],{"className":52907,"style":2943},[2941,2942],[507,52909,52911,52915],{"className":52910},[2570],[507,52912,52825],{"className":52913,"style":52914},[2570,2574],"margin-right:0.0993em;",[507,52916,52918],{"className":52917},[2579],[507,52919,52921,52945],{"className":52920},[2583,3200],[507,52922,52924,52942],{"className":52923},[2587],[507,52925,52927],{"className":52926,"style":4507},[2591],[507,52928,52930,52933],{"style":52929},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[507,52931],{"className":52932,"style":2600},[2599],[507,52934,52936],{"className":52935},[2604,2605,2606,2607],[507,52937,52939],{"className":52938},[2570,2607],[507,52940,41704],{"className":52941,"style":41776},[2570,2611,2607],[507,52943,3225],{"className":52944},[3224],[507,52946,52948],{"className":52947},[2587],[507,52949,52951],{"className":52950,"style":3232},[2591],[507,52952],{},[507,52954],{"className":52955,"style":2965},[2714],[507,52957,52959,52962,53002,53005,53008,53011,53014,53017,53020],{"className":52958},[2937],[507,52960,2810],{"className":52961,"style":2943},[2941,2942],[507,52963,52965,52968],{"className":52964},[2570],[507,52966,3174],{"className":52967,"style":3220},[2570,2611],[507,52969,52971],{"className":52970},[2579],[507,52972,52974,52994],{"className":52973},[2583,3200],[507,52975,52977,52991],{"className":52976},[2587],[507,52978,52980],{"className":52979,"style":4507},[2591],[507,52981,52982,52985],{"style":7637},[507,52983],{"className":52984,"style":2600},[2599],[507,52986,52988],{"className":52987},[2604,2605,2606,2607],[507,52989,3286],{"className":52990},[2570,2611,2607],[507,52992,3225],{"className":52993},[3224],[507,52995,52997],{"className":52996},[2587],[507,52998,53000],{"className":52999,"style":3232},[2591],[507,53001],{},[507,53003,580],{"className":53004},[2941],[507,53006,41704],{"className":53007,"style":41776},[2570,2611],[507,53009,2819],{"className":53010},[2961],[507,53012],{"className":53013,"style":2965},[2714],[507,53015,22278],{"className":53016,"style":27338},[2570,2611],[507,53018,3649],{"className":53019},[2780],[507,53021,2872],{"className":53022,"style":2943},[2780,2942],[507,53024,2749],{"className":53025,"style":2943},[2780,2942],[507,53027],{"className":53028,"style":2965},[2714],[507,53030,53],{"className":53031},[2570],[18,53033,53034],{},"This probability expression contains a squared sinusoid. Using the identity",[507,53036,53038],{"className":53037},[2784],[507,53039,53041,53117],{"className":53040},[2523],[507,53042,53044],{"className":53043},[2527],[2529,53045,53046],{"xmlns":2531,"display":2793},[2533,53047,53048,53114],{},[2536,53049,53050,53060,53080,53082,53088],{},[2539,53051,53052,53058],{},[2536,53053,53054,53056],{},[2542,53055,46721],{},[2689,53057,36690],{},[2693,53059,584],{},[2536,53061,53062,53064,53078],{},[2689,53063,580],{"fence":2557},[9491,53065,53066,53076],{},[2536,53067,53068,53074],{},[3168,53069,53070,53072],{},[2542,53071,42662],{"mathvariant":2748},[2542,53073,20370],{},[2542,53075,41704],{},[2693,53077,584],{},[2689,53079,3649],{"fence":2557},[2689,53081,573],{},[9491,53083,53084,53086],{},[2693,53085,625],{},[2693,53087,584],{},[2536,53089,53090,53092,53094,53096,53098,53100,53102,53108,53110,53112],{},[2689,53091,12248],{"fence":2557},[2693,53093,625],{},[2689,53095,2691],{},[2542,53097,37932],{},[2689,53099,36690],{},[2689,53101,580],{"stretchy":2755},[3168,53103,53104,53106],{},[2542,53105,42662],{"mathvariant":2748},[2542,53107,20370],{},[2542,53109,41704],{},[2689,53111,3649],{"stretchy":2755},[2689,53113,12273],{"fence":2557},[2549,53115,53116],{"encoding":2551},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[507,53118,53120,53284],{"className":53119,"ariaHidden":2557},[2556],[507,53121,53123,53126,53155,53158,53275,53278,53281],{"className":53122},[2561],[507,53124],{"className":53125,"style":37668},[2565],[507,53127,53129,53132],{"className":53128},[7570],[507,53130,46721],{"className":53131},[7570],[507,53133,53135],{"className":53134},[2579],[507,53136,53138],{"className":53137},[2583],[507,53139,53141],{"className":53140},[2587],[507,53142,53144],{"className":53143,"style":46997},[2591],[507,53145,53146,53149],{"style":47000},[507,53147],{"className":53148,"style":2600},[2599],[507,53150,53152],{"className":53151},[2604,2605,2606,2607],[507,53153,584],{"className":53154},[2570,2607],[507,53156],{"className":53157,"style":2965},[2714],[507,53159,53161,53167,53269],{"className":53160},[2937],[507,53162,53164],{"className":53163,"style":2943},[2941,2942],[507,53165,580],{"className":53166},[2947,2606],[507,53168,53170,53173,53266],{"className":53169},[2570],[507,53171],{"className":53172},[2941,9793],[507,53174,53176],{"className":53175},[9491],[507,53177,53179,53258],{"className":53178},[2583,3200],[507,53180,53182,53255],{"className":53181},[2587],[507,53183,53185,53196,53204],{"className":53184,"style":47040},[2591],[507,53186,53187,53190],{"style":11717},[507,53188],{"className":53189,"style":4310},[2599],[507,53191,53193],{"className":53192},[2570],[507,53194,584],{"className":53195},[2570],[507,53197,53198,53201],{"style":9878},[507,53199],{"className":53200,"style":4310},[2599],[507,53202],{"className":53203,"style":9886},[9885],[507,53205,53206,53209],{"style":9889},[507,53207],{"className":53208,"style":4310},[2599],[507,53210,53212,53252],{"className":53211},[2570],[507,53213,53215,53218],{"className":53214},[2570],[507,53216,42662],{"className":53217},[2570],[507,53219,53221],{"className":53220},[2579],[507,53222,53224,53244],{"className":53223},[2583,3200],[507,53225,53227,53241],{"className":53226},[2587],[507,53228,53230],{"className":53229,"style":3207},[2591],[507,53231,53232,53235],{"style":5398},[507,53233],{"className":53234,"style":2600},[2599],[507,53236,53238],{"className":53237},[2604,2605,2606,2607],[507,53239,20370],{"className":53240,"style":20395},[2570,2611,2607],[507,53242,3225],{"className":53243},[3224],[507,53245,53247],{"className":53246},[2587],[507,53248,53250],{"className":53249,"style":3232},[2591],[507,53251],{},[507,53253,41704],{"className":53254,"style":41776},[2570,2611],[507,53256,3225],{"className":53257},[3224],[507,53259,53261],{"className":53260},[2587],[507,53262,53264],{"className":53263,"style":11755},[2591],[507,53265],{},[507,53267],{"className":53268},[2780,9793],[507,53270,53272],{"className":53271,"style":2943},[2780,2942],[507,53273,3649],{"className":53274},[2947,2606],[507,53276],{"className":53277,"style":2919},[2714],[507,53279,573],{"className":53280},[2923],[507,53282],{"className":53283,"style":2919},[2714],[507,53285,53287,53290,53352,53355],{"className":53286},[2561],[507,53288],{"className":53289,"style":48803},[2565],[507,53291,53293,53296,53349],{"className":53292},[2570],[507,53294],{"className":53295},[2941,9793],[507,53297,53299],{"className":53298},[9491],[507,53300,53302,53341],{"className":53301},[2583,3200],[507,53303,53305,53338],{"className":53304},[2587],[507,53306,53308,53319,53327],{"className":53307,"style":9806},[2591],[507,53309,53310,53313],{"style":11717},[507,53311],{"className":53312,"style":4310},[2599],[507,53314,53316],{"className":53315},[2570],[507,53317,584],{"className":53318},[2570],[507,53320,53321,53324],{"style":9878},[507,53322],{"className":53323,"style":4310},[2599],[507,53325],{"className":53326,"style":9886},[9885],[507,53328,53329,53332],{"style":9889},[507,53330],{"className":53331,"style":4310},[2599],[507,53333,53335],{"className":53334},[2570],[507,53336,625],{"className":53337},[2570],[507,53339,3225],{"className":53340},[3224],[507,53342,53344],{"className":53343},[2587],[507,53345,53347],{"className":53346,"style":11755},[2591],[507,53348],{},[507,53350],{"className":53351},[2780,9793],[507,53353],{"className":53354,"style":2965},[2714],[507,53356,53358,53361,53364,53367,53370,53373,53376,53379,53419,53422,53425],{"className":53357},[2937],[507,53359,12248],{"className":53360,"style":2943},[2941,2942],[507,53362,625],{"className":53363},[2570],[507,53365],{"className":53366,"style":2715},[2714],[507,53368,2691],{"className":53369},[2719],[507,53371],{"className":53372,"style":2715},[2714],[507,53374,37932],{"className":53375},[7570],[507,53377,580],{"className":53378},[2941],[507,53380,53382,53385],{"className":53381},[2570],[507,53383,42662],{"className":53384},[2570],[507,53386,53388],{"className":53387},[2579],[507,53389,53391,53411],{"className":53390},[2583,3200],[507,53392,53394,53408],{"className":53393},[2587],[507,53395,53397],{"className":53396,"style":3207},[2591],[507,53398,53399,53402],{"style":5398},[507,53400],{"className":53401,"style":2600},[2599],[507,53403,53405],{"className":53404},[2604,2605,2606,2607],[507,53406,20370],{"className":53407,"style":20395},[2570,2611,2607],[507,53409,3225],{"className":53410},[3224],[507,53412,53414],{"className":53413},[2587],[507,53415,53417],{"className":53416,"style":3232},[2591],[507,53418],{},[507,53420,41704],{"className":53421,"style":41776},[2570,2611],[507,53423,3649],{"className":53424},[2780],[507,53426,12273],{"className":53427,"style":2943},[2780,2942],[18,53429,53430],{},"shows that the dominant oscillatory spectral component appears at the generalized frequency",[507,53432,53434],{"className":53433},[2784],[507,53435,53437,53473],{"className":53436},[2523],[507,53438,53440],{"className":53439},[2527],[2529,53441,53442],{"xmlns":2531,"display":2793},[2533,53443,53444,53470],{},[2536,53445,53446,53452,53454,53468],{},[3168,53447,53448,53450],{},[2542,53449,42975],{},[2542,53451,20370],{},[2689,53453,573],{},[9491,53455,53456,53462],{},[3168,53457,53458,53460],{},[2542,53459,42662],{"mathvariant":2748},[2542,53461,20370],{},[2536,53463,53464,53466],{},[2693,53465,584],{},[2542,53467,8563],{},[2542,53469,53],{"mathvariant":2748},[2549,53471,53472],{"encoding":2551},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[507,53474,53476,53531],{"className":53475,"ariaHidden":2557},[2556],[507,53477,53479,53482,53522,53525,53528],{"className":53478},[2561],[507,53480],{"className":53481,"style":25197},[2565],[507,53483,53485,53488],{"className":53484},[2570],[507,53486,42975],{"className":53487,"style":42996},[2570,2611],[507,53489,53491],{"className":53490},[2579],[507,53492,53494,53514],{"className":53493},[2583,3200],[507,53495,53497,53511],{"className":53496},[2587],[507,53498,53500],{"className":53499,"style":3207},[2591],[507,53501,53502,53505],{"style":43011},[507,53503],{"className":53504,"style":2600},[2599],[507,53506,53508],{"className":53507},[2604,2605,2606,2607],[507,53509,20370],{"className":53510,"style":20395},[2570,2611,2607],[507,53512,3225],{"className":53513},[3224],[507,53515,53517],{"className":53516},[2587],[507,53518,53520],{"className":53519,"style":3232},[2591],[507,53521],{},[507,53523],{"className":53524,"style":2919},[2714],[507,53526,573],{"className":53527},[2923],[507,53529],{"className":53530,"style":2919},[2714],[507,53532,53534,53537,53639],{"className":53533},[2561],[507,53535],{"className":53536,"style":48433},[2565],[507,53538,53540,53543,53636],{"className":53539},[2570],[507,53541],{"className":53542},[2941,9793],[507,53544,53546],{"className":53545},[9491],[507,53547,53549,53628],{"className":53548},[2583,3200],[507,53550,53552,53625],{"className":53551},[2587],[507,53553,53555,53569,53577],{"className":53554,"style":47040},[2591],[507,53556,53557,53560],{"style":11717},[507,53558],{"className":53559,"style":4310},[2599],[507,53561,53563,53566],{"className":53562},[2570],[507,53564,584],{"className":53565},[2570],[507,53567,8563],{"className":53568,"style":2776},[2570,2611],[507,53570,53571,53574],{"style":9878},[507,53572],{"className":53573,"style":4310},[2599],[507,53575],{"className":53576,"style":9886},[9885],[507,53578,53579,53582],{"style":9889},[507,53580],{"className":53581,"style":4310},[2599],[507,53583,53585],{"className":53584},[2570],[507,53586,53588,53591],{"className":53587},[2570],[507,53589,42662],{"className":53590},[2570],[507,53592,53594],{"className":53593},[2579],[507,53595,53597,53617],{"className":53596},[2583,3200],[507,53598,53600,53614],{"className":53599},[2587],[507,53601,53603],{"className":53602,"style":3207},[2591],[507,53604,53605,53608],{"style":5398},[507,53606],{"className":53607,"style":2600},[2599],[507,53609,53611],{"className":53610},[2604,2605,2606,2607],[507,53612,20370],{"className":53613,"style":20395},[2570,2611,2607],[507,53615,3225],{"className":53616},[3224],[507,53618,53620],{"className":53619},[2587],[507,53621,53623],{"className":53622,"style":3232},[2591],[507,53624],{},[507,53626,3225],{"className":53627},[3224],[507,53629,53631],{"className":53630},[2587],[507,53632,53634],{"className":53633,"style":11755},[2591],[507,53635],{},[507,53637],{"className":53638},[2780,9793],[507,53640,53],{"className":53641},[2570],[18,53643,53644],{},"The computational workflow subtracts the mean signal along the pulse-duration axis, applies a Hann window to reduce spectral leakage, uses zero-padding to smooth the displayed Fourier grid, and overlays the exact generalized Rabi ridge on the Fourier heatmap. Zero-padding improves visual interpolation of the plotted spectrum, although the physical frequency resolution remains set by the sampled temporal window.",[13,53646,53648],{"id":53647},"computational-methodology","Computational Methodology",[18,53650,53651,53652,53680,53681,53709,53710,10799,53738,53766,53767,53901,53902,54121],{},"The notebook uses a deterministic closed-form workflow, so it does not require numerical time stepping of the Schrödinger equation. It defines a drive-frequency grid ",[507,53653,53655,53668],{"className":53654},[2523],[507,53656,53658],{"className":53657},[2527],[2529,53659,53660],{"xmlns":2531},[2533,53661,53662,53666],{},[2536,53663,53664],{},[2542,53665,22278],{},[2549,53667,22278],{"encoding":2551},[507,53669,53671],{"className":53670,"ariaHidden":2557},[2556],[507,53672,53674,53677],{"className":53673},[2561],[507,53675],{"className":53676,"style":7035},[2565],[507,53678,22278],{"className":53679,"style":27338},[2570,2611]," in GHz and a pulse-duration grid ",[507,53682,53684,53697],{"className":53683},[2523],[507,53685,53687],{"className":53686},[2527],[2529,53688,53689],{"xmlns":2531},[2533,53690,53691,53695],{},[2536,53692,53693],{},[2542,53694,41704],{},[2549,53696,41830],{"encoding":2551},[507,53698,53700],{"className":53699,"ariaHidden":2557},[2556],[507,53701,53703,53706],{"className":53702},[2561],[507,53704],{"className":53705,"style":2639},[2565],[507,53707,41704],{"className":53708,"style":41776},[2570,2611]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[507,53711,53713,53726],{"className":53712},[2523],[507,53714,53716],{"className":53715},[2527],[2529,53717,53718],{"xmlns":2531},[2533,53719,53720,53724],{},[2536,53721,53722],{},[2542,53723,22278],{},[2549,53725,22278],{"encoding":2551},[507,53727,53729],{"className":53728,"ariaHidden":2557},[2556],[507,53730,53732,53735],{"className":53731},[2561],[507,53733],{"className":53734,"style":7035},[2565],[507,53736,22278],{"className":53737,"style":27338},[2570,2611],[507,53739,53741,53754],{"className":53740},[2523],[507,53742,53744],{"className":53743},[2527],[2529,53745,53746],{"xmlns":2531},[2533,53747,53748,53752],{},[2536,53749,53750],{},[2542,53751,41704],{},[2549,53753,41830],{"encoding":2551},[507,53755,53757],{"className":53756,"ariaHidden":2557},[2556],[507,53758,53760,53763],{"className":53759},[2561],[507,53761],{"className":53762,"style":2639},[2565],[507,53764,41704],{"className":53765,"style":41776},[2570,2611],", the workflow computes the detuning ",[507,53768,53770,53804],{"className":53769},[2523],[507,53771,53773],{"className":53772},[2527],[2529,53774,53775],{"xmlns":2531},[2533,53776,53777,53801],{},[2536,53778,53779,53781,53783,53785,53787,53789,53791,53793,53799],{},[2542,53780,42745],{"mathvariant":2748},[2689,53782,573],{},[2693,53784,584],{},[2542,53786,8563],{},[2689,53788,580],{"stretchy":2755},[2542,53790,22278],{},[2689,53792,2691],{},[3168,53794,53795,53797],{},[2542,53796,22278],{},[2693,53798,601],{},[2689,53800,3649],{"stretchy":2755},[2549,53802,53803],{"encoding":2551},"\\Delta=2\\pi(f-f_0)",[507,53805,53807,53825,53852],{"className":53806,"ariaHidden":2557},[2556],[507,53808,53810,53813,53816,53819,53822],{"className":53809},[2561],[507,53811],{"className":53812,"style":2566},[2565],[507,53814,42745],{"className":53815},[2570],[507,53817],{"className":53818,"style":2919},[2714],[507,53820,573],{"className":53821},[2923],[507,53823],{"className":53824,"style":2919},[2714],[507,53826,53828,53831,53834,53837,53840,53843,53846,53849],{"className":53827},[2561],[507,53829],{"className":53830,"style":2769},[2565],[507,53832,584],{"className":53833},[2570],[507,53835,8563],{"className":53836,"style":2776},[2570,2611],[507,53838,580],{"className":53839},[2941],[507,53841,22278],{"className":53842,"style":27338},[2570,2611],[507,53844],{"className":53845,"style":2715},[2714],[507,53847,2691],{"className":53848},[2719],[507,53850],{"className":53851,"style":2715},[2714],[507,53853,53855,53858,53898],{"className":53854},[2561],[507,53856],{"className":53857,"style":2769},[2565],[507,53859,53861,53864],{"className":53860},[2570],[507,53862,22278],{"className":53863,"style":27338},[2570,2611],[507,53865,53867],{"className":53866},[2579],[507,53868,53870,53890],{"className":53869},[2583,3200],[507,53871,53873,53887],{"className":53872},[2587],[507,53874,53876],{"className":53875,"style":14281},[2591],[507,53877,53878,53881],{"style":42267},[507,53879],{"className":53880,"style":2600},[2599],[507,53882,53884],{"className":53883},[2604,2605,2606,2607],[507,53885,601],{"className":53886},[2570,2607],[507,53888,3225],{"className":53889},[3224],[507,53891,53893],{"className":53892},[2587],[507,53894,53896],{"className":53895,"style":3232},[2591],[507,53897],{},[507,53899,3649],{"className":53900},[2780],", evaluates the generalized Rabi angular frequency ",[507,53903,53905,53942],{"className":53904},[2523],[507,53906,53908],{"className":53907},[2527],[2529,53909,53910],{"xmlns":2531},[2533,53911,53912,53940],{},[2536,53913,53914,53920,53922],{},[3168,53915,53916,53918],{},[2542,53917,42662],{"mathvariant":2748},[2542,53919,20370],{},[2689,53921,573],{},[9496,53923,53924],{},[2536,53925,53926,53932,53934],{},[2539,53927,53928,53930],{},[2542,53929,42662],{"mathvariant":2748},[2693,53931,584],{},[2689,53933,2107],{},[2539,53935,53936,53938],{},[2542,53937,42745],{"mathvariant":2748},[2693,53939,584],{},[2549,53941,47183],{"encoding":2551},[507,53943,53945,54000],{"className":53944,"ariaHidden":2557},[2556],[507,53946,53948,53951,53991,53994,53997],{"className":53947},[2561],[507,53949],{"className":53950,"style":3187},[2565],[507,53952,53954,53957],{"className":53953},[2570],[507,53955,42662],{"className":53956},[2570],[507,53958,53960],{"className":53959},[2579],[507,53961,53963,53983],{"className":53962},[2583,3200],[507,53964,53966,53980],{"className":53965},[2587],[507,53967,53969],{"className":53968,"style":3207},[2591],[507,53970,53971,53974],{"style":5398},[507,53972],{"className":53973,"style":2600},[2599],[507,53975,53977],{"className":53976},[2604,2605,2606,2607],[507,53978,20370],{"className":53979,"style":20395},[2570,2611,2607],[507,53981,3225],{"className":53982},[3224],[507,53984,53986],{"className":53985},[2587],[507,53987,53989],{"className":53988,"style":3232},[2591],[507,53990],{},[507,53992],{"className":53993,"style":2919},[2714],[507,53995,573],{"className":53996},[2923],[507,53998],{"className":53999,"style":2919},[2714],[507,54001,54003,54007],{"className":54002},[2561],[507,54004],{"className":54005,"style":54006},[2565],"height:1.04em;vertical-align:-0.1266em;",[507,54008,54010],{"className":54009},[2570,9819],[507,54011,54013,54112],{"className":54012},[2583,3200],[507,54014,54016,54109],{"className":54015},[2587],[507,54017,54020,54096],{"className":54018,"style":54019},[2591],"height:0.9134em;",[507,54021,54023,54026],{"className":54022,"style":4306},[9833],[507,54024],{"className":54025,"style":4310},[2599],[507,54027,54029,54058,54061,54064,54067],{"className":54028,"style":9840},[2570],[507,54030,54032,54035],{"className":54031},[2570],[507,54033,42662],{"className":54034},[2570],[507,54036,54038],{"className":54037},[2579],[507,54039,54041],{"className":54040},[2583],[507,54042,54044],{"className":54043},[2587],[507,54045,54047],{"className":54046,"style":29409},[2591],[507,54048,54049,54052],{"style":29412},[507,54050],{"className":54051,"style":2600},[2599],[507,54053,54055],{"className":54054},[2604,2605,2606,2607],[507,54056,584],{"className":54057},[2570,2607],[507,54059],{"className":54060,"style":2715},[2714],[507,54062,2107],{"className":54063},[2719],[507,54065],{"className":54066,"style":2715},[2714],[507,54068,54070,54073],{"className":54069},[2570],[507,54071,42745],{"className":54072},[2570],[507,54074,54076],{"className":54075},[2579],[507,54077,54079],{"className":54078},[2583],[507,54080,54082],{"className":54081},[2587],[507,54083,54085],{"className":54084,"style":29409},[2591],[507,54086,54087,54090],{"style":29412},[507,54088],{"className":54089,"style":2600},[2599],[507,54091,54093],{"className":54092},[2604,2605,2606,2607],[507,54094,584],{"className":54095},[2570,2607],[507,54097,54099,54102],{"style":54098},"top:-2.8734em;",[507,54100],{"className":54101,"style":4310},[2599],[507,54103,54105],{"className":54104,"style":9854},[9853],[6281,54106,54107],{"xmlns":6283,"width":9857,"height":9858,"viewBox":9859,"preserveAspectRatio":9860},[6290,54108],{"d":9863},[507,54110,3225],{"className":54111},[3224],[507,54113,54115],{"className":54114},[2587],[507,54116,54119],{"className":54117,"style":54118},[2591],"height:0.1266em;",[507,54120],{},", and calculates the analytic excited-state probability",[507,54123,54125],{"className":54124},[2784],[507,54126,54128,54206],{"className":54127},[2523],[507,54129,54131],{"className":54130},[2527],[2529,54132,54133],{"xmlns":2531,"display":2793},[2533,54134,54135,54203],{},[2536,54136,54137,54143,54145,54147,54149,54151,54153,54155,54171,54181,54201],{},[3168,54138,54139,54141],{},[2542,54140,3174],{},[2542,54142,3286],{},[2689,54144,580],{"stretchy":2755},[2542,54146,22278],{},[2689,54148,2819],{"separator":2557},[2542,54150,41704],{},[2689,54152,3649],{"stretchy":2755},[2689,54154,573],{},[9491,54156,54157,54163],{},[2539,54158,54159,54161],{},[2542,54160,42662],{"mathvariant":2748},[2693,54162,584],{},[3775,54164,54165,54167,54169],{},[2542,54166,42662],{"mathvariant":2748},[2542,54168,20370],{},[2693,54170,584],{},[2539,54172,54173,54179],{},[2536,54174,54175,54177],{},[2542,54176,46721],{},[2689,54178,36690],{},[2693,54180,584],{},[2536,54182,54183,54185,54199],{},[2689,54184,580],{"fence":2557},[9491,54186,54187,54197],{},[2536,54188,54189,54195],{},[3168,54190,54191,54193],{},[2542,54192,42662],{"mathvariant":2748},[2542,54194,20370],{},[2542,54196,41704],{},[2693,54198,584],{},[2689,54200,3649],{"fence":2557},[2542,54202,53],{"mathvariant":2748},[2549,54204,54205],{"encoding":2551},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[507,54207,54209,54282],{"className":54208,"ariaHidden":2557},[2556],[507,54210,54212,54215,54255,54258,54261,54264,54267,54270,54273,54276,54279],{"className":54211},[2561],[507,54213],{"className":54214,"style":2769},[2565],[507,54216,54218,54221],{"className":54217},[2570],[507,54219,3174],{"className":54220,"style":3220},[2570,2611],[507,54222,54224],{"className":54223},[2579],[507,54225,54227,54247],{"className":54226},[2583,3200],[507,54228,54230,54244],{"className":54229},[2587],[507,54231,54233],{"className":54232,"style":4507},[2591],[507,54234,54235,54238],{"style":7637},[507,54236],{"className":54237,"style":2600},[2599],[507,54239,54241],{"className":54240},[2604,2605,2606,2607],[507,54242,3286],{"className":54243},[2570,2611,2607],[507,54245,3225],{"className":54246},[3224],[507,54248,54250],{"className":54249},[2587],[507,54251,54253],{"className":54252,"style":3232},[2591],[507,54254],{},[507,54256,580],{"className":54257},[2941],[507,54259,22278],{"className":54260,"style":27338},[2570,2611],[507,54262,2819],{"className":54263},[2961],[507,54265],{"className":54266,"style":2965},[2714],[507,54268,41704],{"className":54269,"style":41776},[2570,2611],[507,54271,3649],{"className":54272},[2780],[507,54274],{"className":54275,"style":2919},[2714],[507,54277,573],{"className":54278},[2923],[507,54280],{"className":54281,"style":2919},[2714],[507,54283,54285,54288,54424,54427,54456,54459,54576,54579],{"className":54284},[2561],[507,54286],{"className":54287,"style":46833},[2565],[507,54289,54291,54294,54421],{"className":54290},[2570],[507,54292],{"className":54293},[2941,9793],[507,54295,54297],{"className":54296},[9491],[507,54298,54300,54413],{"className":54299},[2583,3200],[507,54301,54303,54410],{"className":54302},[2587],[507,54304,54306,54365,54373],{"className":54305,"style":46852},[2591],[507,54307,54308,54311],{"style":11717},[507,54309],{"className":54310,"style":4310},[2599],[507,54312,54314],{"className":54313},[2570],[507,54315,54317,54320],{"className":54316},[2570],[507,54318,42662],{"className":54319},[2570],[507,54321,54323],{"className":54322},[2579],[507,54324,54326,54357],{"className":54325},[2583,3200],[507,54327,54329,54354],{"className":54328},[2587],[507,54330,54332,54343],{"className":54331,"style":46879},[2591],[507,54333,54334,54337],{"style":46882},[507,54335],{"className":54336,"style":2600},[2599],[507,54338,54340],{"className":54339},[2604,2605,2606,2607],[507,54341,20370],{"className":54342,"style":20395},[2570,2611,2607],[507,54344,54345,54348],{"style":46894},[507,54346],{"className":54347,"style":2600},[2599],[507,54349,54351],{"className":54350},[2604,2605,2606,2607],[507,54352,584],{"className":54353},[2570,2607],[507,54355,3225],{"className":54356},[3224],[507,54358,54360],{"className":54359},[2587],[507,54361,54363],{"className":54362,"style":46913},[2591],[507,54364],{},[507,54366,54367,54370],{"style":9878},[507,54368],{"className":54369,"style":4310},[2599],[507,54371],{"className":54372,"style":9886},[9885],[507,54374,54375,54378],{"style":9889},[507,54376],{"className":54377,"style":4310},[2599],[507,54379,54381],{"className":54380},[2570],[507,54382,54384,54387],{"className":54383},[2570],[507,54385,42662],{"className":54386},[2570],[507,54388,54390],{"className":54389},[2579],[507,54391,54393],{"className":54392},[2583],[507,54394,54396],{"className":54395},[2587],[507,54397,54399],{"className":54398,"style":13224},[2591],[507,54400,54401,54404],{"style":2595},[507,54402],{"className":54403,"style":2600},[2599],[507,54405,54407],{"className":54406},[2604,2605,2606,2607],[507,54408,584],{"className":54409},[2570,2607],[507,54411,3225],{"className":54412},[3224],[507,54414,54416],{"className":54415},[2587],[507,54417,54419],{"className":54418,"style":46970},[2591],[507,54420],{},[507,54422],{"className":54423},[2780,9793],[507,54425],{"className":54426,"style":2965},[2714],[507,54428,54430,54433],{"className":54429},[7570],[507,54431,46721],{"className":54432},[7570],[507,54434,54436],{"className":54435},[2579],[507,54437,54439],{"className":54438},[2583],[507,54440,54442],{"className":54441},[2587],[507,54443,54445],{"className":54444,"style":46997},[2591],[507,54446,54447,54450],{"style":47000},[507,54448],{"className":54449,"style":2600},[2599],[507,54451,54453],{"className":54452},[2604,2605,2606,2607],[507,54454,584],{"className":54455},[2570,2607],[507,54457],{"className":54458,"style":2965},[2714],[507,54460,54462,54468,54570],{"className":54461},[2937],[507,54463,54465],{"className":54464,"style":2943},[2941,2942],[507,54466,580],{"className":54467},[2947,2606],[507,54469,54471,54474,54567],{"className":54470},[2570],[507,54472],{"className":54473},[2941,9793],[507,54475,54477],{"className":54476},[9491],[507,54478,54480,54559],{"className":54479},[2583,3200],[507,54481,54483,54556],{"className":54482},[2587],[507,54484,54486,54497,54505],{"className":54485,"style":47040},[2591],[507,54487,54488,54491],{"style":11717},[507,54489],{"className":54490,"style":4310},[2599],[507,54492,54494],{"className":54493},[2570],[507,54495,584],{"className":54496},[2570],[507,54498,54499,54502],{"style":9878},[507,54500],{"className":54501,"style":4310},[2599],[507,54503],{"className":54504,"style":9886},[9885],[507,54506,54507,54510],{"style":9889},[507,54508],{"className":54509,"style":4310},[2599],[507,54511,54513,54553],{"className":54512},[2570],[507,54514,54516,54519],{"className":54515},[2570],[507,54517,42662],{"className":54518},[2570],[507,54520,54522],{"className":54521},[2579],[507,54523,54525,54545],{"className":54524},[2583,3200],[507,54526,54528,54542],{"className":54527},[2587],[507,54529,54531],{"className":54530,"style":3207},[2591],[507,54532,54533,54536],{"style":5398},[507,54534],{"className":54535,"style":2600},[2599],[507,54537,54539],{"className":54538},[2604,2605,2606,2607],[507,54540,20370],{"className":54541,"style":20395},[2570,2611,2607],[507,54543,3225],{"className":54544},[3224],[507,54546,54548],{"className":54547},[2587],[507,54549,54551],{"className":54550,"style":3232},[2591],[507,54552],{},[507,54554,41704],{"className":54555,"style":41776},[2570,2611],[507,54557,3225],{"className":54558},[3224],[507,54560,54562],{"className":54561},[2587],[507,54563,54565],{"className":54564,"style":11755},[2591],[507,54566],{},[507,54568],{"className":54569},[2780,9793],[507,54571,54573],{"className":54572,"style":2943},[2780,2942],[507,54574,3649],{"className":54575},[2947,2606],[507,54577],{"className":54578,"style":2965},[2714],[507,54580,53],{"className":54581},[2570],[18,54583,54584,54585,54700],{},"The routine optionally applies the damping envelope ",[507,54586,54588,54622],{"className":54587},[2523],[507,54589,54591],{"className":54590},[2527],[2529,54592,54593],{"xmlns":2531},[2533,54594,54595,54619],{},[2536,54596,54597,54599,54601,54603,54605,54607,54609,54617],{},[2542,54598,24069],{},[2689,54600,36690],{},[2689,54602,580],{"stretchy":2755},[2689,54604,2691],{},[2542,54606,41704],{},[2542,54608,645],{"mathvariant":2748},[3775,54610,54611,54613,54615],{},[2542,54612,37046],{},[2693,54614,584],{},[2689,54616,20769],{},[2689,54618,3649],{"stretchy":2755},[2549,54620,54621],{"encoding":2551},"\\exp(-\\tau\u002FT_2^\\ast)",[507,54623,54625],{"className":54624,"ariaHidden":2557},[2556],[507,54626,54628,54631,54634,54637,54640,54643,54646,54697],{"className":54627},[2561],[507,54629],{"className":54630,"style":2769},[2565],[507,54632,24069],{"className":54633},[7570],[507,54635,580],{"className":54636},[2941],[507,54638,2691],{"className":54639},[2570],[507,54641,41704],{"className":54642,"style":41776},[2570,2611],[507,54644,645],{"className":54645},[2570],[507,54647,54649,54652],{"className":54648},[2570],[507,54650,37046],{"className":54651,"style":3220},[2570,2611],[507,54653,54655],{"className":54654},[2579],[507,54656,54658,54689],{"className":54657},[2583,3200],[507,54659,54661,54686],{"className":54660},[2587],[507,54662,54664,54675],{"className":54663,"style":36582},[2591],[507,54665,54666,54669],{"style":43272},[507,54667],{"className":54668,"style":2600},[2599],[507,54670,54672],{"className":54671},[2604,2605,2606,2607],[507,54673,584],{"className":54674},[2570,2607],[507,54676,54677,54680],{"style":2595},[507,54678],{"className":54679,"style":2600},[2599],[507,54681,54683],{"className":54682},[2604,2605,2606,2607],[507,54684,20769],{"className":54685},[2719,2607],[507,54687,3225],{"className":54688},[3224],[507,54690,54692],{"className":54691},[2587],[507,54693,54695],{"className":54694,"style":43302},[2591],[507,54696],{},[507,54698,3649],{"className":54699},[2780]," and clips the result to the physical interval",[507,54702,54704],{"className":54703},[2784],[507,54705,54707,54733],{"className":54706},[2523],[507,54708,54710],{"className":54709},[2527],[2529,54711,54712],{"xmlns":2531,"display":2793},[2533,54713,54714,54730],{},[2536,54715,54716,54718,54720,54726,54728],{},[2693,54717,601],{},[2689,54719,27509],{},[3168,54721,54722,54724],{},[2542,54723,3174],{},[2542,54725,3286],{},[2689,54727,27509],{},[2693,54729,44343],{},[2549,54731,54732],{"encoding":2551},"0\\le P_e\\le 1.",[507,54734,54736,54755,54810],{"className":54735,"ariaHidden":2557},[2556],[507,54737,54739,54743,54746,54749,54752],{"className":54738},[2561],[507,54740],{"className":54741,"style":54742},[2565],"height:0.7804em;vertical-align:-0.136em;",[507,54744,601],{"className":54745},[2570],[507,54747],{"className":54748,"style":2919},[2714],[507,54750,27509],{"className":54751},[2923],[507,54753],{"className":54754,"style":2919},[2714],[507,54756,54758,54761,54801,54804,54807],{"className":54757},[2561],[507,54759],{"className":54760,"style":3187},[2565],[507,54762,54764,54767],{"className":54763},[2570],[507,54765,3174],{"className":54766,"style":3220},[2570,2611],[507,54768,54770],{"className":54769},[2579],[507,54771,54773,54793],{"className":54772},[2583,3200],[507,54774,54776,54790],{"className":54775},[2587],[507,54777,54779],{"className":54778,"style":4507},[2591],[507,54780,54781,54784],{"style":7637},[507,54782],{"className":54783,"style":2600},[2599],[507,54785,54787],{"className":54786},[2604,2605,2606,2607],[507,54788,3286],{"className":54789},[2570,2611,2607],[507,54791,3225],{"className":54792},[3224],[507,54794,54796],{"className":54795},[2587],[507,54797,54799],{"className":54798,"style":3232},[2591],[507,54800],{},[507,54802],{"className":54803,"style":2919},[2714],[507,54805,27509],{"className":54806},[2923],[507,54808],{"className":54809,"style":2919},[2714],[507,54811,54813,54816],{"className":54812},[2561],[507,54814],{"className":54815,"style":2729},[2565],[507,54817,44343],{"className":54818},[2570],[18,54820,54821],{},"Finally, the output is rendered as heatmaps, three-dimensional surfaces, one-dimensional slices, and Fourier-domain spectra. Standard scientific Python libraries provide the numerical arrays, visualization routines, and optional interactive rendering tools.",[13,54823,54825],{"id":54824},"graphical-interpretation","Graphical Interpretation",[18,54827,54828,54829,54927],{},"The ",[507,54830,54832,54860],{"className":54831},[2523],[507,54833,54835],{"className":54834},[2527],[2529,54836,54837],{"xmlns":2531},[2533,54838,54839,54857],{},[2536,54840,54841,54847,54849,54851,54853,54855],{},[3168,54842,54843,54845],{},[2542,54844,3174],{},[2542,54846,3286],{},[2689,54848,580],{"stretchy":2755},[2542,54850,22278],{},[2689,54852,2819],{"separator":2557},[2542,54854,41704],{},[2689,54856,3649],{"stretchy":2755},[2549,54858,54859],{"encoding":2551},"P_e(f,\\tau)",[507,54861,54863],{"className":54862,"ariaHidden":2557},[2556],[507,54864,54866,54869,54909,54912,54915,54918,54921,54924],{"className":54865},[2561],[507,54867],{"className":54868,"style":2769},[2565],[507,54870,54872,54875],{"className":54871},[2570],[507,54873,3174],{"className":54874,"style":3220},[2570,2611],[507,54876,54878],{"className":54877},[2579],[507,54879,54881,54901],{"className":54880},[2583,3200],[507,54882,54884,54898],{"className":54883},[2587],[507,54885,54887],{"className":54886,"style":4507},[2591],[507,54888,54889,54892],{"style":7637},[507,54890],{"className":54891,"style":2600},[2599],[507,54893,54895],{"className":54894},[2604,2605,2606,2607],[507,54896,3286],{"className":54897},[2570,2611,2607],[507,54899,3225],{"className":54900},[3224],[507,54902,54904],{"className":54903},[2587],[507,54905,54907],{"className":54906,"style":3232},[2591],[507,54908],{},[507,54910,580],{"className":54911},[2941],[507,54913,22278],{"className":54914,"style":27338},[2570,2611],[507,54916,2819],{"className":54917},[2961],[507,54919],{"className":54920,"style":2965},[2714],[507,54922,41704],{"className":54923,"style":41776},[2570,2611],[507,54925,3649],{"className":54926},[2780]," heatmap shows the predicted probability of detecting the qubit in the excited state after applying a control pulse. Its horizontal axis gives the applied drive frequency. The vertical axis gives the pulse duration. Brighter regions indicate higher excited-state probability.",[18,54929,54930],{},"The resonant interaction region appears near",[507,54932,54934],{"className":54933},[2784],[507,54935,54937,54960],{"className":54936},[2523],[507,54938,54940],{"className":54939},[2527],[2529,54941,54942],{"xmlns":2531,"display":2793},[2533,54943,54944,54958],{},[2536,54945,54946,54948,54950,54956],{},[2542,54947,22278],{},[2689,54949,573],{},[3168,54951,54952,54954],{},[2542,54953,22278],{},[2693,54955,601],{},[2542,54957,53],{"mathvariant":2748},[2549,54959,44039],{"encoding":2551},[507,54961,54963,54981],{"className":54962,"ariaHidden":2557},[2556],[507,54964,54966,54969,54972,54975,54978],{"className":54965},[2561],[507,54967],{"className":54968,"style":7035},[2565],[507,54970,22278],{"className":54971,"style":27338},[2570,2611],[507,54973],{"className":54974,"style":2919},[2714],[507,54976,573],{"className":54977},[2923],[507,54979],{"className":54980,"style":2919},[2714],[507,54982,54984,54987,55027],{"className":54983},[2561],[507,54985],{"className":54986,"style":7035},[2565],[507,54988,54990,54993],{"className":54989},[2570],[507,54991,22278],{"className":54992,"style":27338},[2570,2611],[507,54994,54996],{"className":54995},[2579],[507,54997,54999,55019],{"className":54998},[2583,3200],[507,55000,55002,55016],{"className":55001},[2587],[507,55003,55005],{"className":55004,"style":14281},[2591],[507,55006,55007,55010],{"style":42267},[507,55008],{"className":55009,"style":2600},[2599],[507,55011,55013],{"className":55012},[2604,2605,2606,2607],[507,55014,601],{"className":55015},[2570,2607],[507,55017,3225],{"className":55018},[3224],[507,55020,55022],{"className":55021},[2587],[507,55023,55025],{"className":55024,"style":3232},[2591],[507,55026],{},[507,55028,53],{"className":55029},[2570],[18,55031,55032,55033,55112],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[507,55034,55036,55070],{"className":55035},[2523],[507,55037,55039],{"className":55038},[2527],[2529,55040,55041],{"xmlns":2531},[2533,55042,55043,55067],{},[2536,55044,55045,55047,55049,55051,55053,55055,55057,55059],{},[2542,55046,42662],{"mathvariant":2748},[2542,55048,645],{"mathvariant":2748},[2693,55050,584],{},[2542,55052,8563],{},[2689,55054,573],{},[2693,55056,48407],{},[6167,55058,6169],{},[2536,55060,55061,55063,55065],{},[2542,55062,48414],{"mathvariant":2748},[2542,55064,3138],{"mathvariant":2748},[2542,55066,666],{"mathvariant":2748},[2549,55068,55069],{"encoding":2551},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[507,55071,55073,55094],{"className":55072,"ariaHidden":2557},[2556],[507,55074,55076,55079,55082,55085,55088,55091],{"className":55075},[2561],[507,55077],{"className":55078,"style":2769},[2565],[507,55080,42720],{"className":55081},[2570],[507,55083,8563],{"className":55084,"style":2776},[2570,2611],[507,55086],{"className":55087,"style":2919},[2714],[507,55089,573],{"className":55090},[2923],[507,55092],{"className":55093,"style":2919},[2714],[507,55095,55097,55100,55103,55106],{"className":55096},[2561],[507,55098],{"className":55099,"style":2566},[2565],[507,55101,48407],{"className":55102},[2570],[507,55104,6169],{"className":55105},[2714,43889],[507,55107,55109],{"className":55108},[2570],[507,55110,48527],{"className":55111},[2570,43896],", the ideal resonant maxima appear near",[507,55114,55116],{"className":55115},[2784],[507,55117,55119,55192],{"className":55118},[2523],[507,55120,55122],{"className":55121},[2527],[2529,55123,55124],{"xmlns":2531,"display":2793},[2533,55125,55126,55189],{},[2536,55127,55128,55130,55132,55134,55136,55142,55144,55146,55149,55151,55157,55159,55161,55164,55166,55172,55174,55176,55179,55181,55187],{},[2542,55129,41704],{},[2689,55131,573],{},[2693,55133,48588],{},[6167,55135,6169],{},[2536,55137,55138,55140],{},[2542,55139,4420],{"mathvariant":2748},[2542,55141,14533],{"mathvariant":2748},[2689,55143,2819],{"separator":2557},[6167,55145,6169],{},[2693,55147,55148],{},"75",[6167,55150,6169],{},[2536,55152,55153,55155],{},[2542,55154,4420],{"mathvariant":2748},[2542,55156,14533],{"mathvariant":2748},[2689,55158,2819],{"separator":2557},[6167,55160,6169],{},[2693,55162,55163],{},"125",[6167,55165,6169],{},[2536,55167,55168,55170],{},[2542,55169,4420],{"mathvariant":2748},[2542,55171,14533],{"mathvariant":2748},[2689,55173,2819],{"separator":2557},[6167,55175,6169],{},[2693,55177,55178],{},"175",[6167,55180,6169],{},[2536,55182,55183,55185],{},[2542,55184,4420],{"mathvariant":2748},[2542,55186,14533],{"mathvariant":2748},[2689,55188,2819],{"separator":2557},[2549,55190,55191],{"encoding":2551},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[507,55193,55195,55213],{"className":55194,"ariaHidden":2557},[2556],[507,55196,55198,55201,55204,55207,55210],{"className":55197},[2561],[507,55199],{"className":55200,"style":2639},[2565],[507,55202,41704],{"className":55203,"style":41776},[2570,2611],[507,55205],{"className":55206,"style":2919},[2714],[507,55208,573],{"className":55209},[2923],[507,55211],{"className":55212,"style":2919},[2714],[507,55214,55216,55220,55223,55226,55232,55235,55238,55241,55244,55247,55253,55256,55259,55262,55265,55268,55274,55277,55280,55283,55286,55289,55295],{"className":55215},[2561],[507,55217],{"className":55218,"style":55219},[2565],"height:0.8389em;vertical-align:-0.1944em;",[507,55221,48588],{"className":55222},[2570],[507,55224,6169],{"className":55225},[2714,43889],[507,55227,55229],{"className":55228},[2570],[507,55230,48678],{"className":55231},[2570,43896],[507,55233,2819],{"className":55234},[2961],[507,55236,6169],{"className":55237},[2714],[507,55239],{"className":55240,"style":2965},[2714],[507,55242,55148],{"className":55243},[2570],[507,55245,6169],{"className":55246},[2714,43889],[507,55248,55250],{"className":55249},[2570],[507,55251,48678],{"className":55252},[2570,43896],[507,55254,2819],{"className":55255},[2961],[507,55257,6169],{"className":55258},[2714],[507,55260],{"className":55261,"style":2965},[2714],[507,55263,55163],{"className":55264},[2570],[507,55266,6169],{"className":55267},[2714,43889],[507,55269,55271],{"className":55270},[2570],[507,55272,48678],{"className":55273},[2570,43896],[507,55275,2819],{"className":55276},[2961],[507,55278,6169],{"className":55279},[2714],[507,55281],{"className":55282,"style":2965},[2714],[507,55284,55178],{"className":55285},[2570],[507,55287,6169],{"className":55288},[2714,43889],[507,55290,55292],{"className":55291},[2570],[507,55293,48678],{"className":55294},[2570,43896],[507,55296,2819],{"className":55297},[2961],[18,55299,55300],{},"before the optional phenomenological damping factor is applied.",[18,55302,55303,55304,55373,55374,55500],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[507,55305,55307,55324],{"className":55306},[2523],[507,55308,55310],{"className":55309},[2527],[2529,55311,55312],{"xmlns":2531},[2533,55313,55314,55322],{},[2536,55315,55316],{},[3168,55317,55318,55320],{},[2542,55319,42662],{"mathvariant":2748},[2542,55321,20370],{},[2549,55323,42902],{"encoding":2551},[507,55325,55327],{"className":55326,"ariaHidden":2557},[2556],[507,55328,55330,55333],{"className":55329},[2561],[507,55331],{"className":55332,"style":3187},[2565],[507,55334,55336,55339],{"className":55335},[2570],[507,55337,42662],{"className":55338},[2570],[507,55340,55342],{"className":55341},[2579],[507,55343,55345,55365],{"className":55344},[2583,3200],[507,55346,55348,55362],{"className":55347},[2587],[507,55349,55351],{"className":55350,"style":3207},[2591],[507,55352,55353,55356],{"style":5398},[507,55354],{"className":55355,"style":2600},[2599],[507,55357,55359],{"className":55358},[2604,2605,2606,2607],[507,55360,20370],{"className":55361,"style":20395},[2570,2611,2607],[507,55363,3225],{"className":55364},[3224],[507,55366,55368],{"className":55367},[2587],[507,55369,55371],{"className":55370,"style":3232},[2591],[507,55372],{},", which increases with detuning, and from the amplitude coefficient ",[507,55375,55377,55405],{"className":55376},[2523],[507,55378,55380],{"className":55379},[2527],[2529,55381,55382],{"xmlns":2531},[2533,55383,55384,55402],{},[2536,55385,55386,55392,55394],{},[2539,55387,55388,55390],{},[2542,55389,42662],{"mathvariant":2748},[2693,55391,584],{},[2542,55393,645],{"mathvariant":2748},[3775,55395,55396,55398,55400],{},[2542,55397,42662],{"mathvariant":2748},[2542,55399,20370],{},[2693,55401,584],{},[2549,55403,55404],{"encoding":2551},"\\Omega^2\u002F\\Omega_R^2",[507,55406,55408],{"className":55407,"ariaHidden":2557},[2556],[507,55409,55411,55415,55444,55447],{"className":55410},[2561],[507,55412],{"className":55413,"style":55414},[2565],"height:1.0894em;vertical-align:-0.2753em;",[507,55416,55418,55421],{"className":55417},[2570],[507,55419,42662],{"className":55420},[2570],[507,55422,55424],{"className":55423},[2579],[507,55425,55427],{"className":55426},[2583],[507,55428,55430],{"className":55429},[2587],[507,55431,55433],{"className":55432,"style":13224},[2591],[507,55434,55435,55438],{"style":2595},[507,55436],{"className":55437,"style":2600},[2599],[507,55439,55441],{"className":55440},[2604,2605,2606,2607],[507,55442,584],{"className":55443},[2570,2607],[507,55445,645],{"className":55446},[2570],[507,55448,55450,55453],{"className":55449},[2570],[507,55451,42662],{"className":55452},[2570],[507,55454,55456],{"className":55455},[2579],[507,55457,55459,55491],{"className":55458},[2583,3200],[507,55460,55462,55488],{"className":55461},[2587],[507,55463,55465,55477],{"className":55464,"style":13224},[2591],[507,55466,55468,55471],{"style":55467},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[507,55469],{"className":55470,"style":2600},[2599],[507,55472,55474],{"className":55473},[2604,2605,2606,2607],[507,55475,20370],{"className":55476,"style":20395},[2570,2611,2607],[507,55478,55479,55482],{"style":2595},[507,55480],{"className":55481,"style":2600},[2599],[507,55483,55485],{"className":55484},[2604,2605,2606,2607],[507,55486,584],{"className":55487},[2570,2607],[507,55489,3225],{"className":55490},[3224],[507,55492,55494],{"className":55493},[2587],[507,55495,55498],{"className":55496,"style":55497},[2591],"height:0.2753em;",[507,55499],{},", which decreases with detuning.",[18,55502,55503],{},"The three-dimensional surface plot represents the same probability data as a height field. This view emphasizes the central resonant ridge and the suppressed off-resonant oscillations. Cross-section plots isolate one-dimensional behavior. A fixed-frequency slice shows how probability evolves with pulse duration. At a fixed duration, a complementary slice shows how probability changes across the drive-frequency sweep.",[18,55505,55506,55507,55604],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[507,55508,55510,55537],{"className":55509},[2523],[507,55511,55513],{"className":55512},[2527],[2529,55514,55515],{"xmlns":2531},[2533,55516,55517,55535],{},[2536,55518,55519,55525,55527,55529,55531,55533],{},[3168,55520,55521,55523],{},[2542,55522,3174],{},[2542,55524,3286],{},[2689,55526,580],{"stretchy":2755},[2542,55528,41704],{},[2689,55530,2819],{"separator":2557},[2542,55532,22278],{},[2689,55534,3649],{"stretchy":2755},[2549,55536,52719],{"encoding":2551},[507,55538,55540],{"className":55539,"ariaHidden":2557},[2556],[507,55541,55543,55546,55586,55589,55592,55595,55598,55601],{"className":55542},[2561],[507,55544],{"className":55545,"style":2769},[2565],[507,55547,55549,55552],{"className":55548},[2570],[507,55550,3174],{"className":55551,"style":3220},[2570,2611],[507,55553,55555],{"className":55554},[2579],[507,55556,55558,55578],{"className":55557},[2583,3200],[507,55559,55561,55575],{"className":55560},[2587],[507,55562,55564],{"className":55563,"style":4507},[2591],[507,55565,55566,55569],{"style":7637},[507,55567],{"className":55568,"style":2600},[2599],[507,55570,55572],{"className":55571},[2604,2605,2606,2607],[507,55573,3286],{"className":55574},[2570,2611,2607],[507,55576,3225],{"className":55577},[3224],[507,55579,55581],{"className":55580},[2587],[507,55582,55584],{"className":55583,"style":3232},[2591],[507,55585],{},[507,55587,580],{"className":55588},[2941],[507,55590,41704],{"className":55591,"style":41776},[2570,2611],[507,55593,2819],{"className":55594},[2961],[507,55596],{"className":55597,"style":2965},[2714],[507,55599,22278],{"className":55600,"style":27338},[2570,2611],[507,55602,3649],{"className":55603},[2780],". An overlaid theoretical curve follows",[507,55606,55608],{"className":55607},[2784],[507,55609,55611,55683],{"className":55610},[2523],[507,55612,55614],{"className":55613},[2527],[2529,55615,55616],{"xmlns":2531,"display":2793},[2533,55617,55618,55680],{},[2536,55619,55620,55626,55628,55630,55632,55634,55678],{},[3168,55621,55622,55624],{},[2542,55623,42975],{},[2542,55625,20370],{},[2689,55627,580],{"stretchy":2755},[2542,55629,22278],{},[2689,55631,3649],{"stretchy":2755},[2689,55633,573],{},[9496,55635,55636],{},[2536,55637,55638,55658,55660,55662,55664,55666,55672],{},[2539,55639,55640,55656],{},[2536,55641,55642,55644,55654],{},[2689,55643,580],{"fence":2557},[9491,55645,55646,55648],{},[2542,55647,42662],{"mathvariant":2748},[2536,55649,55650,55652],{},[2693,55651,584],{},[2542,55653,8563],{},[2689,55655,3649],{"fence":2557},[2693,55657,584],{},[2689,55659,2107],{},[2689,55661,580],{"stretchy":2755},[2542,55663,22278],{},[2689,55665,2691],{},[3168,55667,55668,55670],{},[2542,55669,22278],{},[2693,55671,601],{},[2539,55673,55674,55676],{},[2689,55675,3649],{"stretchy":2755},[2693,55677,584],{},[2542,55679,53],{"mathvariant":2748},[2549,55681,55682],{"encoding":2551},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[507,55684,55686,55750],{"className":55685,"ariaHidden":2557},[2556],[507,55687,55689,55692,55732,55735,55738,55741,55744,55747],{"className":55688},[2561],[507,55690],{"className":55691,"style":2769},[2565],[507,55693,55695,55698],{"className":55694},[2570],[507,55696,42975],{"className":55697,"style":42996},[2570,2611],[507,55699,55701],{"className":55700},[2579],[507,55702,55704,55724],{"className":55703},[2583,3200],[507,55705,55707,55721],{"className":55706},[2587],[507,55708,55710],{"className":55709,"style":3207},[2591],[507,55711,55712,55715],{"style":43011},[507,55713],{"className":55714,"style":2600},[2599],[507,55716,55718],{"className":55717},[2604,2605,2606,2607],[507,55719,20370],{"className":55720,"style":20395},[2570,2611,2607],[507,55722,3225],{"className":55723},[3224],[507,55725,55727],{"className":55726},[2587],[507,55728,55730],{"className":55729,"style":3232},[2591],[507,55731],{},[507,55733,580],{"className":55734},[2941],[507,55736,22278],{"className":55737,"style":27338},[2570,2611],[507,55739,3649],{"className":55740},[2780],[507,55742],{"className":55743,"style":2919},[2714],[507,55745,573],{"className":55746},[2923],[507,55748],{"className":55749,"style":2919},[2714],[507,55751,55753,55756,55999],{"className":55752},[2561],[507,55754],{"className":55755,"style":50025},[2565],[507,55757,55759],{"className":55758},[2570,9819],[507,55760,55762,55991],{"className":55761},[2583,3200],[507,55763,55765,55988],{"className":55764},[2587],[507,55766,55768,55976],{"className":55767,"style":50038},[2591],[507,55769,55771,55774],{"className":55770,"style":50042},[9833],[507,55772],{"className":55773,"style":50046},[2599],[507,55775,55777,55883,55886,55889,55892,55895,55898,55901,55904,55907,55947],{"className":55776,"style":47273},[2570],[507,55778,55780,55860],{"className":55779},[2937],[507,55781,55783,55789,55854],{"className":55782},[2937],[507,55784,55786],{"className":55785,"style":2943},[2941,2942],[507,55787,580],{"className":55788},[2947,2606],[507,55790,55792,55795,55851],{"className":55791},[2570],[507,55793],{"className":55794},[2941,9793],[507,55796,55798],{"className":55797},[9491],[507,55799,55801,55843],{"className":55800},[2583,3200],[507,55802,55804,55840],{"className":55803},[2587],[507,55805,55807,55821,55829],{"className":55806,"style":47040},[2591],[507,55808,55809,55812],{"style":11717},[507,55810],{"className":55811,"style":4310},[2599],[507,55813,55815,55818],{"className":55814},[2570],[507,55816,584],{"className":55817},[2570],[507,55819,8563],{"className":55820,"style":2776},[2570,2611],[507,55822,55823,55826],{"style":9878},[507,55824],{"className":55825,"style":4310},[2599],[507,55827],{"className":55828,"style":9886},[9885],[507,55830,55831,55834],{"style":9889},[507,55832],{"className":55833,"style":4310},[2599],[507,55835,55837],{"className":55836},[2570],[507,55838,42662],{"className":55839},[2570],[507,55841,3225],{"className":55842},[3224],[507,55844,55846],{"className":55845},[2587],[507,55847,55849],{"className":55848,"style":11755},[2591],[507,55850],{},[507,55852],{"className":55853},[2780,9793],[507,55855,55857],{"className":55856,"style":2943},[2780,2942],[507,55858,3649],{"className":55859},[2947,2606],[507,55861,55863],{"className":55862},[2579],[507,55864,55866],{"className":55865},[2583],[507,55867,55869],{"className":55868},[2587],[507,55870,55872],{"className":55871,"style":50145},[2591],[507,55873,55874,55877],{"style":50148},[507,55875],{"className":55876,"style":2600},[2599],[507,55878,55880],{"className":55879},[2604,2605,2606,2607],[507,55881,584],{"className":55882},[2570,2607],[507,55884],{"className":55885,"style":2715},[2714],[507,55887,2107],{"className":55888},[2719],[507,55890],{"className":55891,"style":2715},[2714],[507,55893,580],{"className":55894},[2941],[507,55896,22278],{"className":55897,"style":27338},[2570,2611],[507,55899],{"className":55900,"style":2715},[2714],[507,55902,2691],{"className":55903},[2719],[507,55905],{"className":55906,"style":2715},[2714],[507,55908,55910,55913],{"className":55909},[2570],[507,55911,22278],{"className":55912,"style":27338},[2570,2611],[507,55914,55916],{"className":55915},[2579],[507,55917,55919,55939],{"className":55918},[2583,3200],[507,55920,55922,55936],{"className":55921},[2587],[507,55923,55925],{"className":55924,"style":14281},[2591],[507,55926,55927,55930],{"style":42267},[507,55928],{"className":55929,"style":2600},[2599],[507,55931,55933],{"className":55932},[2604,2605,2606,2607],[507,55934,601],{"className":55935},[2570,2607],[507,55937,3225],{"className":55938},[3224],[507,55940,55942],{"className":55941},[2587],[507,55943,55945],{"className":55944,"style":3232},[2591],[507,55946],{},[507,55948,55950,55953],{"className":55949},[2780],[507,55951,3649],{"className":55952},[2780],[507,55954,55956],{"className":55955},[2579],[507,55957,55959],{"className":55958},[2583],[507,55960,55962],{"className":55961},[2587],[507,55963,55965],{"className":55964,"style":29409},[2591],[507,55966,55967,55970],{"style":29412},[507,55968],{"className":55969,"style":2600},[2599],[507,55971,55973],{"className":55972},[2604,2605,2606,2607],[507,55974,584],{"className":55975},[2570,2607],[507,55977,55978,55981],{"style":50253},[507,55979],{"className":55980,"style":50046},[2599],[507,55982,55984],{"className":55983,"style":50260},[9853],[6281,55985,55986],{"xmlns":6283,"width":9857,"height":50263,"viewBox":50264,"preserveAspectRatio":9860},[6290,55987],{"d":50267},[507,55989,3225],{"className":55990},[3224],[507,55992,55994],{"className":55993},[2587],[507,55995,55997],{"className":55996,"style":50277},[2591],[507,55998],{},[507,56000,53],{"className":56001},[2570],[13,56003,56005],{"id":56004},"primary-modeling-assumptions","Primary Modeling Assumptions",[18,56007,56008],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[41852,56010,56011,56021],{},[41855,56012,56013],{},[41858,56014,56015,56018],{},[41861,56016,56017],{},"Assumption",[41861,56019,56020],{},"Implication",[41868,56022,56023,56107,56115,56122,56130,56167,56257,56414,56462,56470,56478,56486],{},[41858,56024,56025,56028],{},[41873,56026,56027],{},"Two-level truncation",[41873,56029,56030,56031,10799,56069,53],{},"The simulated model includes only ",[507,56032,56034,56051],{"className":56033},[2523],[507,56035,56037],{"className":56036},[2527],[2529,56038,56039],{"xmlns":2531},[2533,56040,56041,56049],{},[2536,56042,56043,56045,56047],{},[2689,56044,2749],{"stretchy":2755},[2542,56046,37913],{},[2689,56048,2756],{"stretchy":2755},[2549,56050,41895],{"encoding":2551},[507,56052,56054],{"className":56053,"ariaHidden":2557},[2556],[507,56055,56057,56060,56063,56066],{"className":56056},[2561],[507,56058],{"className":56059,"style":2769},[2565],[507,56061,2749],{"className":56062},[2941],[507,56064,37913],{"className":56065,"style":2776},[2570,2611],[507,56067,2756],{"className":56068},[2780],[507,56070,56072,56089],{"className":56071},[2523],[507,56073,56075],{"className":56074},[2527],[2529,56076,56077],{"xmlns":2531},[2533,56078,56079,56087],{},[2536,56080,56081,56083,56085],{},[2689,56082,2749],{"stretchy":2755},[2542,56084,3286],{},[2689,56086,2756],{"stretchy":2755},[2549,56088,41941],{"encoding":2551},[507,56090,56092],{"className":56091,"ariaHidden":2557},[2556],[507,56093,56095,56098,56101,56104],{"className":56094},[2561],[507,56096],{"className":56097,"style":2769},[2565],[507,56099,2749],{"className":56100},[2941],[507,56102,3286],{"className":56103},[2570,2611],[507,56105,2756],{"className":56106},[2780],[41858,56108,56109,56112],{},[41873,56110,56111],{},"Classical drive",[41873,56113,56114],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[41858,56116,56117,56119],{},[41873,56118,43315],{},[41873,56120,56121],{},"Fast counter-rotating terms are omitted.",[41858,56123,56124,56127],{},[41873,56125,56126],{},"Square pulse",[41873,56128,56129],{},"The drive is assumed to turn on and off instantaneously.",[41858,56131,56132,56135],{},[41873,56133,56134],{},"Uniform drive amplitude",[41873,56136,56137,56138,56166],{},"The parameter ",[507,56139,56141,56154],{"className":56140},[2523],[507,56142,56144],{"className":56143},[2527],[2529,56145,56146],{"xmlns":2531},[2533,56147,56148,56152],{},[2536,56149,56150],{},[2542,56151,42662],{"mathvariant":2748},[2549,56153,42665],{"encoding":2551},[507,56155,56157],{"className":56156,"ariaHidden":2557},[2556],[507,56158,56160,56163],{"className":56159},[2561],[507,56161],{"className":56162,"style":2566},[2565],[507,56164,42662],{"className":56165},[2570]," remains constant across the frequency sweep.",[41858,56168,56169,56172],{},[41873,56170,56171],{},"Optional damping envelope",[41873,56173,56137,56174,56256],{},[507,56175,56177,56196],{"className":56176},[2523],[507,56178,56180],{"className":56179},[2527],[2529,56181,56182],{"xmlns":2531},[2533,56183,56184,56194],{},[2536,56185,56186],{},[3775,56187,56188,56190,56192],{},[2542,56189,37046],{},[2693,56191,584],{},[2689,56193,20769],{},[2549,56195,43241],{"encoding":2551},[507,56197,56199],{"className":56198,"ariaHidden":2557},[2556],[507,56200,56202,56205],{"className":56201},[2561],[507,56203],{"className":56204,"style":43251},[2565],[507,56206,56208,56211],{"className":56207},[2570],[507,56209,37046],{"className":56210,"style":3220},[2570,2611],[507,56212,56214],{"className":56213},[2579],[507,56215,56217,56248],{"className":56216},[2583,3200],[507,56218,56220,56245],{"className":56219},[2587],[507,56221,56223,56234],{"className":56222,"style":36582},[2591],[507,56224,56225,56228],{"style":43272},[507,56226],{"className":56227,"style":2600},[2599],[507,56229,56231],{"className":56230},[2604,2605,2606,2607],[507,56232,584],{"className":56233},[2570,2607],[507,56235,56236,56239],{"style":2595},[507,56237],{"className":56238,"style":2600},[2599],[507,56240,56242],{"className":56241},[2604,2605,2606,2607],[507,56243,20769],{"className":56244},[2719,2607],[507,56246,3225],{"className":56247},[3224],[507,56249,56251],{"className":56250},[2587],[507,56252,56254],{"className":56253,"style":43302},[2591],[507,56255],{}," enters as a simple exponential contrast envelope.",[41858,56258,56259,56333],{},[41873,56260,56261,56262,56332],{},"No explicit ",[507,56263,56265,56283],{"className":56264},[2523],[507,56266,56268],{"className":56267},[2527],[2529,56269,56270],{"xmlns":2531},[2533,56271,56272,56280],{},[2536,56273,56274],{},[3168,56275,56276,56278],{},[2542,56277,37046],{},[2693,56279,625],{},[2549,56281,56282],{"encoding":2551},"T_1",[507,56284,56286],{"className":56285,"ariaHidden":2557},[2556],[507,56287,56289,56292],{"className":56288},[2561],[507,56290],{"className":56291,"style":3187},[2565],[507,56293,56295,56298],{"className":56294},[2570],[507,56296,37046],{"className":56297,"style":3220},[2570,2611],[507,56299,56301],{"className":56300},[2579],[507,56302,56304,56324],{"className":56303},[2583,3200],[507,56305,56307,56321],{"className":56306},[2587],[507,56308,56310],{"className":56309,"style":14281},[2591],[507,56311,56312,56315],{"style":7637},[507,56313],{"className":56314,"style":2600},[2599],[507,56316,56318],{"className":56317},[2604,2605,2606,2607],[507,56319,625],{"className":56320},[2570,2607],[507,56322,3225],{"className":56323},[3224],[507,56325,56327],{"className":56326},[2587],[507,56328,56330],{"className":56329,"style":3232},[2591],[507,56331],{}," relaxation",[41873,56334,56335,56336,56374,56375,56413],{},"Energy decay from ",[507,56337,56339,56356],{"className":56338},[2523],[507,56340,56342],{"className":56341},[2527],[2529,56343,56344],{"xmlns":2531},[2533,56345,56346,56354],{},[2536,56347,56348,56350,56352],{},[2689,56349,2749],{"stretchy":2755},[2542,56351,3286],{},[2689,56353,2756],{"stretchy":2755},[2549,56355,41941],{"encoding":2551},[507,56357,56359],{"className":56358,"ariaHidden":2557},[2556],[507,56360,56362,56365,56368,56371],{"className":56361},[2561],[507,56363],{"className":56364,"style":2769},[2565],[507,56366,2749],{"className":56367},[2941],[507,56369,3286],{"className":56370},[2570,2611],[507,56372,2756],{"className":56373},[2780]," to ",[507,56376,56378,56395],{"className":56377},[2523],[507,56379,56381],{"className":56380},[2527],[2529,56382,56383],{"xmlns":2531},[2533,56384,56385,56393],{},[2536,56386,56387,56389,56391],{},[2689,56388,2749],{"stretchy":2755},[2542,56390,37913],{},[2689,56392,2756],{"stretchy":2755},[2549,56394,41895],{"encoding":2551},[507,56396,56398],{"className":56397,"ariaHidden":2557},[2556],[507,56399,56401,56404,56407,56410],{"className":56400},[2561],[507,56402],{"className":56403,"style":2769},[2565],[507,56405,2749],{"className":56406},[2941],[507,56408,37913],{"className":56409,"style":2776},[2570,2611],[507,56411,2756],{"className":56412},[2780]," is omitted.",[41858,56415,56416,56419],{},[41873,56417,56418],{},"No higher-state leakage",[41873,56420,56421,56422,56461],{},"Higher levels, including a transmon ",[507,56423,56425,56443],{"className":56424},[2523],[507,56426,56428],{"className":56427},[2527],[2529,56429,56430],{"xmlns":2531},[2533,56431,56432,56440],{},[2536,56433,56434,56436,56438],{},[2689,56435,2749],{"stretchy":2755},[2542,56437,22278],{},[2689,56439,2756],{"stretchy":2755},[2549,56441,56442],{"encoding":2551},"\\lvert f\\rangle",[507,56444,56446],{"className":56445,"ariaHidden":2557},[2556],[507,56447,56449,56452,56455,56458],{"className":56448},[2561],[507,56450],{"className":56451,"style":2769},[2565],[507,56453,2749],{"className":56454},[2941],[507,56456,22278],{"className":56457,"style":27338},[2570,2611],[507,56459,2756],{"className":56460},[2780]," state, are excluded.",[41858,56463,56464,56467],{},[41873,56465,56466],{},"No AC Stark shift",[41873,56468,56469],{},"Drive-induced transition-frequency shifts are excluded.",[41858,56471,56472,56475],{},[41873,56473,56474],{},"No Bloch-Siegert shift",[41873,56476,56477],{},"Counter-rotating corrections are excluded.",[41858,56479,56480,56483],{},[41873,56481,56482],{},"No experimental readout model",[41873,56484,56485],{},"Assignment errors and finite signal-to-noise effects are omitted.",[41858,56487,56488,56491],{},[41873,56489,56490],{},"No pulse-envelope shaping",[41873,56492,56493],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[18,56495,56496],{},"A hardware-accurate superconducting-qubit simulation requires additional structure, including anharmonic multilevel dynamics, calibrated microwave pulse envelopes, drive phase control, amplitude-dependent frequency shifts, finite thermal populations, leakage channels, and measurement infidelity modeled with density matrix dynamics.",[13,56498,56500],{"id":56499},"notebook-control-knobs","Notebook Control Knobs",[18,56502,56503],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[41852,56505,56506,56519],{},[41855,56507,56508],{},[41858,56509,56510,56513,56517],{},[41861,56511,56512],{},"Parameter",[41861,56514,56516],{"align":56515},"right","Default",[41861,56518,41866],{},[41868,56520,56521,56605,56658,56673,56688,56703,56718,56733,56747],{},[41858,56522,56523,56528,56532],{},[41873,56524,56525],{},[504,56526,56527],{},"F0_GHZ",[41873,56529,56530],{"align":56515},[504,56531,43802],{},[41873,56533,56534,56535,56604],{},"Qubit transition frequency ",[507,56536,56538,56555],{"className":56537},[2523],[507,56539,56541],{"className":56540},[2527],[2529,56542,56543],{"xmlns":2531},[2533,56544,56545,56553],{},[2536,56546,56547],{},[3168,56548,56549,56551],{},[2542,56550,22278],{},[2693,56552,601],{},[2549,56554,42237],{"encoding":2551},[507,56556,56558],{"className":56557,"ariaHidden":2557},[2556],[507,56559,56561,56564],{"className":56560},[2561],[507,56562],{"className":56563,"style":7035},[2565],[507,56565,56567,56570],{"className":56566},[2570],[507,56568,22278],{"className":56569,"style":27338},[2570,2611],[507,56571,56573],{"className":56572},[2579],[507,56574,56576,56596],{"className":56575},[2583,3200],[507,56577,56579,56593],{"className":56578},[2587],[507,56580,56582],{"className":56581,"style":14281},[2591],[507,56583,56584,56587],{"style":42267},[507,56585],{"className":56586,"style":2600},[2599],[507,56588,56590],{"className":56589},[2604,2605,2606,2607],[507,56591,601],{"className":56592},[2570,2607],[507,56594,3225],{"className":56595},[3224],[507,56597,56599],{"className":56598},[2587],[507,56600,56602],{"className":56601,"style":3232},[2591],[507,56603],{}," in GHz",[41858,56606,56607,56612,56617],{},[41873,56608,56609],{},[504,56610,56611],{},"OMEGA_ONRESONANCE_MHZ",[41873,56613,56614],{"align":56515},[504,56615,56616],{},"20.0",[41873,56618,56619,56620,56657],{},"On-resonance Rabi rate ",[507,56621,56623,56642],{"className":56622},[2523],[507,56624,56626],{"className":56625},[2527],[2529,56627,56628],{"xmlns":2531},[2533,56629,56630,56640],{},[2536,56631,56632,56634,56636,56638],{},[2542,56633,42662],{"mathvariant":2748},[2542,56635,645],{"mathvariant":2748},[2693,56637,584],{},[2542,56639,8563],{},[2549,56641,42707],{"encoding":2551},[507,56643,56645],{"className":56644,"ariaHidden":2557},[2556],[507,56646,56648,56651,56654],{"className":56647},[2561],[507,56649],{"className":56650,"style":2769},[2565],[507,56652,42720],{"className":56653},[2570],[507,56655,8563],{"className":56656,"style":2776},[2570,2611]," in MHz",[41858,56659,56660,56665,56670],{},[41873,56661,56662],{},[504,56663,56664],{},"FREQ_MIN_GHZ",[41873,56666,56667],{"align":56515},[504,56668,56669],{},"4.90",[41873,56671,56672],{},"Minimum swept drive frequency in GHz",[41858,56674,56675,56680,56685],{},[41873,56676,56677],{},[504,56678,56679],{},"FREQ_MAX_GHZ",[41873,56681,56682],{"align":56515},[504,56683,56684],{},"5.10",[41873,56686,56687],{},"Maximum swept drive frequency in GHz",[41858,56689,56690,56695,56700],{},[41873,56691,56692],{},[504,56693,56694],{},"N_FREQ",[41873,56696,56697],{"align":56515},[504,56698,56699],{},"401",[41873,56701,56702],{},"Number of drive-frequency samples",[41858,56704,56705,56710,56715],{},[41873,56706,56707],{},[504,56708,56709],{},"DUR_MIN_NS",[41873,56711,56712],{"align":56515},[504,56713,56714],{},"0.0",[41873,56716,56717],{},"Minimum pulse duration in ns",[41858,56719,56720,56725,56730],{},[41873,56721,56722],{},[504,56723,56724],{},"DUR_MAX_NS",[41873,56726,56727],{"align":56515},[504,56728,56729],{},"200.0",[41873,56731,56732],{},"Maximum pulse duration in ns",[41858,56734,56735,56740,56744],{},[41873,56736,56737],{},[504,56738,56739],{},"N_DUR",[41873,56741,56742],{"align":56515},[504,56743,56699],{},[41873,56745,56746],{},"Number of pulse-duration samples",[41858,56748,56749,56754,56759],{},[41873,56750,56751],{},[504,56752,56753],{},"T2STAR_NS",[41873,56755,56756],{"align":56515},[504,56757,56758],{},"500.0",[41873,56760,56761,56762,56765],{},"Optional damping time in ns, with ",[504,56763,56764],{},"None"," disabling damping",[18,56767,56768],{},"Additional cross-section plotting cells expose localized slice controls.",[41852,56770,56771,56781],{},[41855,56772,56773],{},[41858,56774,56775,56777,56779],{},[41861,56776,56512],{},[41861,56778,56516],{"align":56515},[41861,56780,41866],{},[41868,56782,56783,56882],{},[41858,56784,56785,56790,56794],{},[41873,56786,56787],{},[504,56788,56789],{},"F_CROSS_GHZ",[41873,56791,56792],{"align":56515},[504,56793,43802],{},[41873,56795,56796,56797],{},"Drive-frequency slice used to plot ",[507,56798,56800,56824],{"className":56799},[2523],[507,56801,56803],{"className":56802},[2527],[2529,56804,56805],{"xmlns":2531},[2533,56806,56807,56821],{},[2536,56808,56809,56815,56817,56819],{},[3168,56810,56811,56813],{},[2542,56812,3174],{},[2542,56814,3286],{},[2689,56816,580],{"stretchy":2755},[2542,56818,41704],{},[2689,56820,3649],{"stretchy":2755},[2549,56822,56823],{"encoding":2551},"P_e(\\tau)",[507,56825,56827],{"className":56826,"ariaHidden":2557},[2556],[507,56828,56830,56833,56873,56876,56879],{"className":56829},[2561],[507,56831],{"className":56832,"style":2769},[2565],[507,56834,56836,56839],{"className":56835},[2570],[507,56837,3174],{"className":56838,"style":3220},[2570,2611],[507,56840,56842],{"className":56841},[2579],[507,56843,56845,56865],{"className":56844},[2583,3200],[507,56846,56848,56862],{"className":56847},[2587],[507,56849,56851],{"className":56850,"style":4507},[2591],[507,56852,56853,56856],{"style":7637},[507,56854],{"className":56855,"style":2600},[2599],[507,56857,56859],{"className":56858},[2604,2605,2606,2607],[507,56860,3286],{"className":56861},[2570,2611,2607],[507,56863,3225],{"className":56864},[3224],[507,56866,56868],{"className":56867},[2587],[507,56869,56871],{"className":56870,"style":3232},[2591],[507,56872],{},[507,56874,580],{"className":56875},[2941],[507,56877,41704],{"className":56878,"style":41776},[2570,2611],[507,56880,3649],{"className":56881},[2780],[41858,56883,56884,56889,56894],{},[41873,56885,56886],{},[504,56887,56888],{},"T_CROSS_NS",[41873,56890,56891],{"align":56515},[504,56892,56893],{},"100.0",[41873,56895,56896,56897],{},"Pulse-duration slice used to plot ",[507,56898,56900,56924],{"className":56899},[2523],[507,56901,56903],{"className":56902},[2527],[2529,56904,56905],{"xmlns":2531},[2533,56906,56907,56921],{},[2536,56908,56909,56915,56917,56919],{},[3168,56910,56911,56913],{},[2542,56912,3174],{},[2542,56914,3286],{},[2689,56916,580],{"stretchy":2755},[2542,56918,22278],{},[2689,56920,3649],{"stretchy":2755},[2549,56922,56923],{"encoding":2551},"P_e(f)",[507,56925,56927],{"className":56926,"ariaHidden":2557},[2556],[507,56928,56930,56933,56973,56976,56979],{"className":56929},[2561],[507,56931],{"className":56932,"style":2769},[2565],[507,56934,56936,56939],{"className":56935},[2570],[507,56937,3174],{"className":56938,"style":3220},[2570,2611],[507,56940,56942],{"className":56941},[2579],[507,56943,56945,56965],{"className":56944},[2583,3200],[507,56946,56948,56962],{"className":56947},[2587],[507,56949,56951],{"className":56950,"style":4507},[2591],[507,56952,56953,56956],{"style":7637},[507,56954],{"className":56955,"style":2600},[2599],[507,56957,56959],{"className":56958},[2604,2605,2606,2607],[507,56960,3286],{"className":56961},[2570,2611,2607],[507,56963,3225],{"className":56964},[3224],[507,56966,56968],{"className":56967},[2587],[507,56969,56971],{"className":56970,"style":3232},[2591],[507,56972],{},[507,56974,580],{"className":56975},[2941],[507,56977,22278],{"className":56978,"style":27338},[2570,2611],[507,56980,3649],{"className":56981},[2780],[18,56983,56984],{},"Separate FFT processing controls support spectral analysis.",[41852,56986,56987,56997],{},[41855,56988,56989],{},[41858,56990,56991,56993,56995],{},[41861,56992,56512],{},[41861,56994,56516],{"align":56515},[41861,56996,41866],{},[41868,56998,56999,57013,57028,57042,57057,57072],{},[41858,57000,57001,57006,57010],{},[41873,57002,57003],{},[504,57004,57005],{},"FFT_PAD",[41873,57007,57008],{"align":56515},[504,57009,35740],{},[41873,57011,57012],{},"Zero-padding factor used in the Fourier transform",[41858,57014,57015,57020,57025],{},[41873,57016,57017],{},[504,57018,57019],{},"WINDOW",[41873,57021,57022],{"align":56515},[504,57023,57024],{},"\"hann\"",[41873,57026,57027],{},"Window function applied along the pulse-duration axis",[41858,57029,57030,57035,57039],{},[41873,57031,57032],{},[504,57033,57034],{},"DETREND_MEAN",[41873,57036,57037],{"align":56515},[504,57038,13878],{},[41873,57040,57041],{},"Flag that subtracts the mean before the FFT",[41858,57043,57044,57049,57054],{},[41873,57045,57046],{},[504,57047,57048],{},"AMP_MODE",[41873,57050,57051],{"align":56515},[504,57052,57053],{},"\"magnitude\"",[41873,57055,57056],{},"Display mode for magnitude, power, or dB-scaled amplitude",[41858,57058,57059,57064,57069],{},[41873,57060,57061],{},[504,57062,57063],{},"NORM_MODE",[41873,57065,57066],{"align":56515},[504,57067,57068],{},"\"global\"",[41873,57070,57071],{},"Normalization mode applied globally or per drive-frequency column",[41858,57073,57074,57079,57084],{},[41873,57075,57076],{},[504,57077,57078],{},"FREQ_UNITS",[41873,57080,57081],{"align":56515},[504,57082,57083],{},"\"MHz\"",[41873,57085,57086],{},"Frequency units used on the Fourier-frequency axis",[18,57088,57089],{},"Three-dimensional visualization cells add rendering controls for upsampling, colormaps, face-count limits, and viewing angles. These settings change the appearance of the rendered figures while preserving the analytic probability model.",[498,57091,57093],{"className":500,"code":57092,"language":502,"meta":104,"style":104},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[504,57094,57095,57100,57107],{"__ignoreMap":104},[507,57096,57097],{"class":509,"line":510},[507,57098,57099],{"class":562},"# @title Install dependencies\n",[507,57101,57102,57104],{"class":509,"line":105},[507,57103,514],{"class":513},[507,57105,57106],{"class":517}," sys\n",[507,57108,57109,57111,57114,57116],{"class":509,"line":540},[507,57110,8769],{"class":572},[507,57112,57113],{"class":517},"pip ",[507,57115,2367],{"class":572},[507,57117,57118],{"class":517},"q install pyvista pythreejs trame\n",[498,57120,57122],{"className":500,"code":57121,"language":502,"meta":104,"style":104},"# @title Controls & imports\nimport math\nfrom typing import Optional, Tuple\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\ntry:\n    import pyvista as pv  # optional; only for 3D interactive\n    PV_AVAILABLE = True\nexcept Exception:\n    PV_AVAILABLE = False\n\nplt.rcParams['figure.dpi'] = 250\nplt.rcParams['axes.grid'] = False\n\n# ---------- Control knobs ----------\nF0_GHZ: float = 5.000        # Qubit transition freq f0 [GHz]\nOMEGA_ONRESONANCE_MHZ: float = 20.0  # Ω\u002F2π on-resonance [MHz]\nFREQ_MIN_GHZ: float = 4.90   # Sweep min f [GHz]\nFREQ_MAX_GHZ: float = 5.10   # Sweep max f [GHz]\nN_FREQ: int = 401            # # of frequency points\nDUR_MIN_NS: float = 0.0      # Sweep min duration τ [ns]\nDUR_MAX_NS: float = 200.0    # Sweep max duration τ [ns]\nN_DUR: int = 401             # # of duration points\nT2STAR_NS: Optional[float] = 500.0  # None to disable exp(-τ\u002FT2*)\n\nprint('Controls loaded.')\n",[504,57123,57124,57129,57136,57147,57157,57168,57180,57184,57191,57207,57217,57225,57234,57238,57253,57266,57270,57275,57291,57307,57323,57339,57355,57371,57387,57402,57421,57425],{"__ignoreMap":104},[507,57125,57126],{"class":509,"line":510},[507,57127,57128],{"class":562},"# @title Controls & imports\n",[507,57130,57131,57133],{"class":509,"line":105},[507,57132,514],{"class":513},[507,57134,57135],{"class":517}," math\n",[507,57137,57138,57140,57142,57144],{"class":509,"line":540},[507,57139,529],{"class":513},[507,57141,1166],{"class":517},[507,57143,514],{"class":513},[507,57145,57146],{"class":517}," Optional, Tuple\n",[507,57148,57149,57151,57153,57155],{"class":509,"line":553},[507,57150,514],{"class":513},[507,57152,518],{"class":517},[507,57154,521],{"class":513},[507,57156,524],{"class":517},[507,57158,57159,57161,57164,57166],{"class":509,"line":559},[507,57160,514],{"class":513},[507,57162,57163],{"class":517}," matplotlib.pyplot ",[507,57165,521],{"class":513},[507,57167,1159],{"class":517},[507,57169,57170,57172,57175,57177],{"class":509,"line":566},[507,57171,514],{"class":513},[507,57173,57174],{"class":517}," pandas ",[507,57176,521],{"class":513},[507,57178,57179],{"class":517}," pd\n",[507,57181,57182],{"class":509,"line":590},[507,57183,556],{"emptyLinePlaceholder":133},[507,57185,57186,57189],{"class":509,"line":610},[507,57187,57188],{"class":513},"try",[507,57190,1728],{"class":517},[507,57192,57193,57196,57199,57201,57204],{"class":509,"line":634},[507,57194,57195],{"class":513},"    import",[507,57197,57198],{"class":517}," pyvista ",[507,57200,521],{"class":513},[507,57202,57203],{"class":517}," pv  ",[507,57205,57206],{"class":562},"# optional; only for 3D interactive\n",[507,57208,57209,57212,57214],{"class":509,"line":661},[507,57210,57211],{"class":583},"    PV_AVAILABLE",[507,57213,1423],{"class":572},[507,57215,57216],{"class":583}," True\n",[507,57218,57219,57222],{"class":509,"line":678},[507,57220,57221],{"class":513},"except",[507,57223,57224],{"class":517}," Exception:\n",[507,57226,57227,57229,57231],{"class":509,"line":683},[507,57228,57211],{"class":583},[507,57230,1423],{"class":572},[507,57232,57233],{"class":583}," False\n",[507,57235,57236],{"class":509,"line":697},[507,57237,556],{"emptyLinePlaceholder":133},[507,57239,57240,57243,57246,57248,57250],{"class":509,"line":710},[507,57241,57242],{"class":517},"plt.rcParams[",[507,57244,57245],{"class":730},"'figure.dpi'",[507,57247,8206],{"class":517},[507,57249,573],{"class":572},[507,57251,57252],{"class":583}," 250\n",[507,57254,57255,57257,57260,57262,57264],{"class":509,"line":715},[507,57256,57242],{"class":517},[507,57258,57259],{"class":730},"'axes.grid'",[507,57261,8206],{"class":517},[507,57263,573],{"class":572},[507,57265,57233],{"class":583},[507,57267,57268],{"class":509,"line":721},[507,57269,556],{"emptyLinePlaceholder":133},[507,57271,57272],{"class":509,"line":736},[507,57273,57274],{"class":562},"# ---------- Control knobs ----------\n",[507,57276,57277,57279,57281,57283,57285,57288],{"class":509,"line":748},[507,57278,56527],{"class":583},[507,57280,1403],{"class":517},[507,57282,1406],{"class":572},[507,57284,1423],{"class":572},[507,57286,57287],{"class":583}," 5.000",[507,57289,57290],{"class":562},"        # Qubit transition freq f0 [GHz]\n",[507,57292,57293,57295,57297,57299,57301,57304],{"class":509,"line":761},[507,57294,56611],{"class":583},[507,57296,1403],{"class":517},[507,57298,1406],{"class":572},[507,57300,1423],{"class":572},[507,57302,57303],{"class":583}," 20.0",[507,57305,57306],{"class":562},"  # Ω\u002F2π on-resonance [MHz]\n",[507,57308,57309,57311,57313,57315,57317,57320],{"class":509,"line":775},[507,57310,56664],{"class":583},[507,57312,1403],{"class":517},[507,57314,1406],{"class":572},[507,57316,1423],{"class":572},[507,57318,57319],{"class":583}," 4.90",[507,57321,57322],{"class":562},"   # Sweep min f [GHz]\n",[507,57324,57325,57327,57329,57331,57333,57336],{"class":509,"line":784},[507,57326,56679],{"class":583},[507,57328,1403],{"class":517},[507,57330,1406],{"class":572},[507,57332,1423],{"class":572},[507,57334,57335],{"class":583}," 5.10",[507,57337,57338],{"class":562},"   # Sweep max f [GHz]\n",[507,57340,57341,57343,57345,57347,57349,57352],{"class":509,"line":796},[507,57342,56694],{"class":583},[507,57344,1403],{"class":517},[507,57346,1420],{"class":572},[507,57348,1423],{"class":572},[507,57350,57351],{"class":583}," 401",[507,57353,57354],{"class":562},"            # # of frequency points\n",[507,57356,57357,57359,57361,57363,57365,57368],{"class":509,"line":809},[507,57358,56709],{"class":583},[507,57360,1403],{"class":517},[507,57362,1406],{"class":572},[507,57364,1423],{"class":572},[507,57366,57367],{"class":583}," 0.0",[507,57369,57370],{"class":562},"      # Sweep min duration τ [ns]\n",[507,57372,57373,57375,57377,57379,57381,57384],{"class":509,"line":1352},[507,57374,56724],{"class":583},[507,57376,1403],{"class":517},[507,57378,1406],{"class":572},[507,57380,1423],{"class":572},[507,57382,57383],{"class":583}," 200.0",[507,57385,57386],{"class":562},"    # Sweep max duration τ [ns]\n",[507,57388,57389,57391,57393,57395,57397,57399],{"class":509,"line":1357},[507,57390,56739],{"class":583},[507,57392,1403],{"class":517},[507,57394,1420],{"class":572},[507,57396,1423],{"class":572},[507,57398,57351],{"class":583},[507,57400,57401],{"class":562},"             # # of duration points\n",[507,57403,57404,57406,57409,57411,57413,57415,57418],{"class":509,"line":1362},[507,57405,56753],{"class":583},[507,57407,57408],{"class":517},": Optional[",[507,57410,1406],{"class":572},[507,57412,8206],{"class":517},[507,57414,573],{"class":572},[507,57416,57417],{"class":583}," 500.0",[507,57419,57420],{"class":562},"  # None to disable exp(-τ\u002FT2*)\n",[507,57422,57423],{"class":509,"line":1367},[507,57424,556],{"emptyLinePlaceholder":133},[507,57426,57427,57429,57431,57434],{"class":509,"line":1379},[507,57428,8525],{"class":572},[507,57430,580],{"class":517},[507,57432,57433],{"class":730},"'Controls loaded.'",[507,57435,587],{"class":517},[498,57437,57440],{"className":57438,"code":57439,"language":7039,"meta":104},[8531],"Controls loaded.\n",[504,57441,57439],{"__ignoreMap":104},[498,57443,57445],{"className":500,"code":57444,"language":502,"meta":104,"style":104},"def rabi_probability(\n    f_ghz: np.ndarray,\n    tau_ns: np.ndarray,\n    f0_ghz: float,\n    omega_onres_mhz: float,\n    t2star_ns: Optional[float] = None,\n) -> np.ndarray:\n    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n\n    Args:\n        f_ghz: 1D array of drive frequencies [GHz].\n        tau_ns: 1D array of pulse durations [ns].\n        f0_ghz: Qubit frequency on resonance [GHz].\n        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n\n    Returns:\n        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n    \"\"\"\n    two_pi = 2.0 * math.pi\n    f = f_ghz * 1e9\n    f0 = f0_ghz * 1e9\n    tau = tau_ns * 1e-9\n    omega = two_pi * omega_onres_mhz * 1e6  # rad\u002Fs\n\n    F, TAU = np.meshgrid(f, tau, indexing='xy')\n    Delta = two_pi * (F - f0)               # rad\u002Fs\n    Omega_R = np.sqrt(omega**2 + Delta**2)\n\n    with np.errstate(divide='ignore', invalid='ignore'):\n        amp = (omega \u002F Omega_R) ** 2\n        phase = 0.5 * Omega_R * TAU\n        Pe = amp * np.sin(phase) ** 2\n\n    if t2star_ns is not None and t2star_ns > 0.0:\n        decay = np.exp(-TAU \u002F (t2star_ns * 1e-9))\n        Pe = Pe * decay\n\n    return np.clip(Pe, 0.0, 1.0)\n",[504,57446,57447,57456,57463,57470,57481,57492,57510,57515,57520,57524,57528,57533,57538,57543,57548,57553,57557,57561,57566,57570,57585,57600,57614,57629,57652,57656,57684,57706,57735,57739,57770,57790,57810,57833,57837,57861,57891,57905,57909],{"__ignoreMap":104},[507,57448,57449,57451,57454],{"class":509,"line":510},[507,57450,1370],{"class":513},[507,57452,57453],{"class":576}," rabi_probability",[507,57455,1376],{"class":517},[507,57457,57458,57461],{"class":509,"line":105},[507,57459,57460],{"class":1382},"    f_ghz",[507,57462,1386],{"class":517},[507,57464,57465,57468],{"class":509,"line":540},[507,57466,57467],{"class":1382},"    tau_ns",[507,57469,1386],{"class":517},[507,57471,57472,57475,57477,57479],{"class":509,"line":553},[507,57473,57474],{"class":1382},"    f0_ghz",[507,57476,1403],{"class":517},[507,57478,1406],{"class":572},[507,57480,1409],{"class":517},[507,57482,57483,57486,57488,57490],{"class":509,"line":559},[507,57484,57485],{"class":1382},"    omega_onres_mhz",[507,57487,1403],{"class":517},[507,57489,1406],{"class":572},[507,57491,1409],{"class":517},[507,57493,57494,57497,57499,57501,57503,57505,57508],{"class":509,"line":566},[507,57495,57496],{"class":1382},"    t2star_ns",[507,57498,57408],{"class":517},[507,57500,1406],{"class":572},[507,57502,8206],{"class":517},[507,57504,573],{"class":572},[507,57506,57507],{"class":583}," None",[507,57509,1409],{"class":517},[507,57511,57512],{"class":509,"line":590},[507,57513,57514],{"class":517},") -> np.ndarray:\n",[507,57516,57517],{"class":509,"line":610},[507,57518,57519],{"class":730},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[507,57521,57522],{"class":509,"line":634},[507,57523,556],{"emptyLinePlaceholder":133},[507,57525,57526],{"class":509,"line":661},[507,57527,1485],{"class":730},[507,57529,57530],{"class":509,"line":678},[507,57531,57532],{"class":730},"        f_ghz: 1D array of drive frequencies [GHz].\n",[507,57534,57535],{"class":509,"line":683},[507,57536,57537],{"class":730},"        tau_ns: 1D array of pulse durations [ns].\n",[507,57539,57540],{"class":509,"line":697},[507,57541,57542],{"class":730},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[507,57544,57545],{"class":509,"line":710},[507,57546,57547],{"class":730},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[507,57549,57550],{"class":509,"line":715},[507,57551,57552],{"class":730},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[507,57554,57555],{"class":509,"line":721},[507,57556,556],{"emptyLinePlaceholder":133},[507,57558,57559],{"class":509,"line":736},[507,57560,1556],{"class":730},[507,57562,57563],{"class":509,"line":748},[507,57564,57565],{"class":730},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[507,57567,57568],{"class":509,"line":761},[507,57569,1468],{"class":730},[507,57571,57572,57575,57577,57580,57582],{"class":509,"line":775},[507,57573,57574],{"class":517},"    two_pi ",[507,57576,573],{"class":572},[507,57578,57579],{"class":583}," 2.0",[507,57581,8229],{"class":572},[507,57583,57584],{"class":517}," math.pi\n",[507,57586,57587,57590,57592,57595,57597],{"class":509,"line":784},[507,57588,57589],{"class":517},"    f ",[507,57591,573],{"class":572},[507,57593,57594],{"class":517}," f_ghz ",[507,57596,2391],{"class":572},[507,57598,57599],{"class":583}," 1e9\n",[507,57601,57602,57605,57607,57610,57612],{"class":509,"line":796},[507,57603,57604],{"class":517},"    f0 ",[507,57606,573],{"class":572},[507,57608,57609],{"class":517}," f0_ghz ",[507,57611,2391],{"class":572},[507,57613,57599],{"class":583},[507,57615,57616,57619,57621,57624,57626],{"class":509,"line":809},[507,57617,57618],{"class":517},"    tau ",[507,57620,573],{"class":572},[507,57622,57623],{"class":517}," tau_ns ",[507,57625,2391],{"class":572},[507,57627,57628],{"class":583}," 1e-9\n",[507,57630,57631,57634,57636,57639,57641,57644,57646,57649],{"class":509,"line":1352},[507,57632,57633],{"class":517},"    omega ",[507,57635,573],{"class":572},[507,57637,57638],{"class":517}," two_pi ",[507,57640,2391],{"class":572},[507,57642,57643],{"class":517}," omega_onres_mhz ",[507,57645,2391],{"class":572},[507,57647,57648],{"class":583}," 1e6",[507,57650,57651],{"class":562},"  # rad\u002Fs\n",[507,57653,57654],{"class":509,"line":1357},[507,57655,556],{"emptyLinePlaceholder":133},[507,57657,57658,57661,57664,57666,57668,57671,57674,57677,57679,57682],{"class":509,"line":1362},[507,57659,57660],{"class":517},"    F, ",[507,57662,57663],{"class":583},"TAU",[507,57665,1423],{"class":572},[507,57667,1616],{"class":517},[507,57669,57670],{"class":576},"meshgrid",[507,57672,57673],{"class":517},"(f, tau, ",[507,57675,57676],{"class":2155},"indexing",[507,57678,573],{"class":572},[507,57680,57681],{"class":730},"'xy'",[507,57683,587],{"class":517},[507,57685,57686,57689,57691,57693,57695,57698,57700,57703],{"class":509,"line":1367},[507,57687,57688],{"class":517},"    Delta ",[507,57690,573],{"class":572},[507,57692,57638],{"class":517},[507,57694,2391],{"class":572},[507,57696,57697],{"class":517}," (F ",[507,57699,2367],{"class":572},[507,57701,57702],{"class":517}," f0)               ",[507,57704,57705],{"class":562},"# rad\u002Fs\n",[507,57707,57708,57711,57713,57715,57717,57720,57722,57724,57726,57729,57731,57733],{"class":509,"line":1379},[507,57709,57710],{"class":517},"    Omega_R ",[507,57712,573],{"class":572},[507,57714,1616],{"class":517},[507,57716,9819],{"class":576},[507,57718,57719],{"class":517},"(omega",[507,57721,2377],{"class":572},[507,57723,584],{"class":583},[507,57725,8313],{"class":572},[507,57727,57728],{"class":517}," Delta",[507,57730,2377],{"class":572},[507,57732,584],{"class":583},[507,57734,587],{"class":517},[507,57736,57737],{"class":509,"line":1389},[507,57738,556],{"emptyLinePlaceholder":133},[507,57740,57741,57744,57746,57749,57751,57754,57756,57759,57761,57764,57766,57768],{"class":509,"line":1397},[507,57742,57743],{"class":513},"    with",[507,57745,1616],{"class":517},[507,57747,57748],{"class":576},"errstate",[507,57750,580],{"class":517},[507,57752,57753],{"class":2155},"divide",[507,57755,573],{"class":572},[507,57757,57758],{"class":730},"'ignore'",[507,57760,622],{"class":517},[507,57762,57763],{"class":2155},"invalid",[507,57765,573],{"class":572},[507,57767,57758],{"class":730},[507,57769,1883],{"class":517},[507,57771,57772,57775,57777,57780,57782,57785,57787],{"class":509,"line":1412},[507,57773,57774],{"class":517},"        amp ",[507,57776,573],{"class":572},[507,57778,57779],{"class":517}," (omega ",[507,57781,645],{"class":572},[507,57783,57784],{"class":517}," Omega_R) ",[507,57786,2377],{"class":572},[507,57788,57789],{"class":583}," 2\n",[507,57791,57792,57795,57797,57800,57802,57805,57807],{"class":509,"line":1431},[507,57793,57794],{"class":517},"        phase ",[507,57796,573],{"class":572},[507,57798,57799],{"class":583}," 0.5",[507,57801,8229],{"class":572},[507,57803,57804],{"class":517}," Omega_R ",[507,57806,2391],{"class":572},[507,57808,57809],{"class":583}," TAU\n",[507,57811,57812,57815,57817,57820,57822,57824,57826,57829,57831],{"class":509,"line":1449},[507,57813,57814],{"class":517},"        Pe ",[507,57816,573],{"class":572},[507,57818,57819],{"class":517}," amp ",[507,57821,2391],{"class":572},[507,57823,1616],{"class":517},[507,57825,46721],{"class":576},[507,57827,57828],{"class":517},"(phase) ",[507,57830,2377],{"class":572},[507,57832,57789],{"class":583},[507,57834,57835],{"class":509,"line":1465},[507,57836,556],{"emptyLinePlaceholder":133},[507,57838,57839,57841,57844,57846,57848,57850,57853,57855,57857,57859],{"class":509,"line":1471},[507,57840,1717],{"class":513},[507,57842,57843],{"class":517}," t2star_ns ",[507,57845,37008],{"class":513},[507,57847,21980],{"class":513},[507,57849,57507],{"class":583},[507,57851,57852],{"class":513}," and",[507,57854,57843],{"class":517},[507,57856,1651],{"class":572},[507,57858,57367],{"class":583},[507,57860,1728],{"class":517},[507,57862,57863,57866,57868,57870,57872,57874,57876,57878,57881,57884,57886,57889],{"class":509,"line":1477},[507,57864,57865],{"class":517},"        decay ",[507,57867,573],{"class":572},[507,57869,1616],{"class":517},[507,57871,24069],{"class":576},[507,57873,580],{"class":517},[507,57875,2367],{"class":572},[507,57877,57663],{"class":583},[507,57879,57880],{"class":572}," \u002F",[507,57882,57883],{"class":517}," (t2star_ns ",[507,57885,2391],{"class":572},[507,57887,57888],{"class":583}," 1e-9",[507,57890,22540],{"class":517},[507,57892,57893,57895,57897,57900,57902],{"class":509,"line":1482},[507,57894,57814],{"class":517},[507,57896,573],{"class":572},[507,57898,57899],{"class":517}," Pe ",[507,57901,2391],{"class":572},[507,57903,57904],{"class":517}," decay\n",[507,57906,57907],{"class":509,"line":1488},[507,57908,556],{"emptyLinePlaceholder":133},[507,57910,57911,57913,57915,57918,57921,57923,57925,57928],{"class":509,"line":1494},[507,57912,2504],{"class":513},[507,57914,1616],{"class":517},[507,57916,57917],{"class":576},"clip",[507,57919,57920],{"class":517},"(Pe, ",[507,57922,56714],{"class":583},[507,57924,622],{"class":517},[507,57926,57927],{"class":583},"1.0",[507,57929,587],{"class":517},[498,57931,57933],{"className":500,"code":57932,"language":502,"meta":104,"style":104},"freq_ghz = np.linspace(FREQ_MIN_GHZ, FREQ_MAX_GHZ, N_FREQ)\ndur_ns = np.linspace(DUR_MIN_NS, DUR_MAX_NS, N_DUR)\nPe = rabi_probability(freq_ghz, dur_ns, F0_GHZ, OMEGA_ONRESONANCE_MHZ, T2STAR_NS)\nprint('Grid shape (τ × f):', Pe.shape)\n",[504,57934,57935,57961,57986,58010],{"__ignoreMap":104},[507,57936,57937,57940,57942,57944,57947,57949,57951,57953,57955,57957,57959],{"class":509,"line":510},[507,57938,57939],{"class":517},"freq_ghz ",[507,57941,573],{"class":572},[507,57943,1616],{"class":517},[507,57945,57946],{"class":576},"linspace",[507,57948,580],{"class":517},[507,57950,56664],{"class":583},[507,57952,622],{"class":517},[507,57954,56679],{"class":583},[507,57956,622],{"class":517},[507,57958,56694],{"class":583},[507,57960,587],{"class":517},[507,57962,57963,57966,57968,57970,57972,57974,57976,57978,57980,57982,57984],{"class":509,"line":105},[507,57964,57965],{"class":517},"dur_ns ",[507,57967,573],{"class":572},[507,57969,1616],{"class":517},[507,57971,57946],{"class":576},[507,57973,580],{"class":517},[507,57975,56709],{"class":583},[507,57977,622],{"class":517},[507,57979,56724],{"class":583},[507,57981,622],{"class":517},[507,57983,56739],{"class":583},[507,57985,587],{"class":517},[507,57987,57988,57991,57993,57995,57998,58000,58002,58004,58006,58008],{"class":509,"line":540},[507,57989,57990],{"class":517},"Pe ",[507,57992,573],{"class":572},[507,57994,57453],{"class":576},[507,57996,57997],{"class":517},"(freq_ghz, dur_ns, ",[507,57999,56527],{"class":583},[507,58001,622],{"class":517},[507,58003,56611],{"class":583},[507,58005,622],{"class":517},[507,58007,56753],{"class":583},[507,58009,587],{"class":517},[507,58011,58012,58014,58016,58019],{"class":509,"line":553},[507,58013,8525],{"class":572},[507,58015,580],{"class":517},[507,58017,58018],{"class":730},"'Grid shape (τ × f):'",[507,58020,58021],{"class":517},", Pe.shape)\n",[498,58023,58026],{"className":58024,"code":58025,"language":7039,"meta":104},[8531],"Grid shape (τ × f): (401, 401)\n",[504,58027,58025],{"__ignoreMap":104},[498,58029,58031],{"className":500,"code":58030,"language":502,"meta":104,"style":104},"controls_table = pd.DataFrame(\n    [\n        ['f0 (GHz)', F0_GHZ],\n        ['Ω\u002F2π on-res (MHz)', OMEGA_ONRESONANCE_MHZ],\n        ['f sweep min (GHz)', FREQ_MIN_GHZ],\n        ['f sweep max (GHz)', FREQ_MAX_GHZ],\n        ['N_f', N_FREQ],\n        ['τ sweep min (ns)', DUR_MIN_NS],\n        ['τ sweep max (ns)', DUR_MAX_NS],\n        ['N_τ', N_DUR],\n        ['T2* (ns; None=off)', T2STAR_NS],\n    ],\n    columns=['Parameter', 'Value']\n)\ncontrols_table\n",[504,58032,58033,58048,58053,58068,58081,58094,58107,58120,58133,58146,58159,58172,58177,58196,58200],{"__ignoreMap":104},[507,58034,58035,58038,58040,58043,58046],{"class":509,"line":510},[507,58036,58037],{"class":517},"controls_table ",[507,58039,573],{"class":572},[507,58041,58042],{"class":517}," pd.",[507,58044,58045],{"class":576},"DataFrame",[507,58047,1376],{"class":517},[507,58049,58050],{"class":509,"line":105},[507,58051,58052],{"class":517},"    [\n",[507,58054,58055,58058,58061,58063,58065],{"class":509,"line":540},[507,58056,58057],{"class":517},"        [",[507,58059,58060],{"class":730},"'f0 (GHz)'",[507,58062,622],{"class":517},[507,58064,56527],{"class":583},[507,58066,58067],{"class":517},"],\n",[507,58069,58070,58072,58075,58077,58079],{"class":509,"line":553},[507,58071,58057],{"class":517},[507,58073,58074],{"class":730},"'Ω\u002F2π on-res (MHz)'",[507,58076,622],{"class":517},[507,58078,56611],{"class":583},[507,58080,58067],{"class":517},[507,58082,58083,58085,58088,58090,58092],{"class":509,"line":559},[507,58084,58057],{"class":517},[507,58086,58087],{"class":730},"'f sweep min (GHz)'",[507,58089,622],{"class":517},[507,58091,56664],{"class":583},[507,58093,58067],{"class":517},[507,58095,58096,58098,58101,58103,58105],{"class":509,"line":566},[507,58097,58057],{"class":517},[507,58099,58100],{"class":730},"'f sweep max (GHz)'",[507,58102,622],{"class":517},[507,58104,56679],{"class":583},[507,58106,58067],{"class":517},[507,58108,58109,58111,58114,58116,58118],{"class":509,"line":590},[507,58110,58057],{"class":517},[507,58112,58113],{"class":730},"'N_f'",[507,58115,622],{"class":517},[507,58117,56694],{"class":583},[507,58119,58067],{"class":517},[507,58121,58122,58124,58127,58129,58131],{"class":509,"line":610},[507,58123,58057],{"class":517},[507,58125,58126],{"class":730},"'τ sweep min (ns)'",[507,58128,622],{"class":517},[507,58130,56709],{"class":583},[507,58132,58067],{"class":517},[507,58134,58135,58137,58140,58142,58144],{"class":509,"line":634},[507,58136,58057],{"class":517},[507,58138,58139],{"class":730},"'τ sweep max (ns)'",[507,58141,622],{"class":517},[507,58143,56724],{"class":583},[507,58145,58067],{"class":517},[507,58147,58148,58150,58153,58155,58157],{"class":509,"line":661},[507,58149,58057],{"class":517},[507,58151,58152],{"class":730},"'N_τ'",[507,58154,622],{"class":517},[507,58156,56739],{"class":583},[507,58158,58067],{"class":517},[507,58160,58161,58163,58166,58168,58170],{"class":509,"line":678},[507,58162,58057],{"class":517},[507,58164,58165],{"class":730},"'T2* (ns; None=off)'",[507,58167,622],{"class":517},[507,58169,56753],{"class":583},[507,58171,58067],{"class":517},[507,58173,58174],{"class":509,"line":683},[507,58175,58176],{"class":517},"    ],\n",[507,58178,58179,58182,58184,58186,58189,58191,58194],{"class":509,"line":697},[507,58180,58181],{"class":2155},"    columns",[507,58183,573],{"class":572},[507,58185,12248],{"class":517},[507,58187,58188],{"class":730},"'Parameter'",[507,58190,622],{"class":517},[507,58192,58193],{"class":730},"'Value'",[507,58195,1794],{"class":517},[507,58197,58198],{"class":509,"line":710},[507,58199,587],{"class":517},[507,58201,58202],{"class":509,"line":715},[507,58203,58204],{"class":517},"controls_table\n",[498,58206,58209],{"className":58207,"code":58208,"language":7039,"meta":104},[8531],"Parameter  Value\n0            f0 (GHz)    5.0\n1   Ω\u002F2π on-res (MHz)   20.0\n2   f sweep min (GHz)    4.9\n3   f sweep max (GHz)    5.1\n4                 N_f  401.0\n5    τ sweep min (ns)    0.0\n6    τ sweep max (ns)  200.0\n7                 N_τ  401.0\n8  T2* (ns; None=off)  500.0\n",[504,58210,58208],{"__ignoreMap":104},[498,58212,58214],{"className":500,"code":58213,"language":502,"meta":104,"style":104},"fig, ax = plt.subplots(figsize=(7, 5))\nc = ax.pcolormesh(freq_ghz, dur_ns, Pe, shading='auto')\ncb = fig.colorbar(c, ax=ax, label=r'$P_e$')\nax.set_xlabel('Drive frequency f (GHz)')\nax.set_ylabel('Pulse duration τ (ns)')\nax.set_title('Rabi excited-state probability $P_e(f,\\\\,\\\\tau)$')\nfig.tight_layout()\nplt.show()\n",[504,58215,58216,58248,58274,58310,58325,58339,58363,58373],{"__ignoreMap":104},[507,58217,58218,58221,58223,58226,58229,58231,58234,58236,58238,58241,58243,58246],{"class":509,"line":510},[507,58219,58220],{"class":517},"fig, ax ",[507,58222,573],{"class":572},[507,58224,58225],{"class":517}," plt.",[507,58227,58228],{"class":576},"subplots",[507,58230,580],{"class":517},[507,58232,58233],{"class":2155},"figsize",[507,58235,573],{"class":572},[507,58237,580],{"class":517},[507,58239,58240],{"class":583},"7",[507,58242,622],{"class":517},[507,58244,58245],{"class":583},"5",[507,58247,22540],{"class":517},[507,58249,58250,58253,58255,58258,58261,58264,58267,58269,58272],{"class":509,"line":105},[507,58251,58252],{"class":517},"c ",[507,58254,573],{"class":572},[507,58256,58257],{"class":517}," ax.",[507,58259,58260],{"class":576},"pcolormesh",[507,58262,58263],{"class":517},"(freq_ghz, dur_ns, Pe, ",[507,58265,58266],{"class":2155},"shading",[507,58268,573],{"class":572},[507,58270,58271],{"class":730},"'auto'",[507,58273,587],{"class":517},[507,58275,58276,58279,58281,58284,58287,58290,58293,58295,58298,58300,58302,58304,58308],{"class":509,"line":540},[507,58277,58278],{"class":517},"cb ",[507,58280,573],{"class":572},[507,58282,58283],{"class":517}," fig.",[507,58285,58286],{"class":576},"colorbar",[507,58288,58289],{"class":517},"(c, ",[507,58291,58292],{"class":2155},"ax",[507,58294,573],{"class":572},[507,58296,58297],{"class":517},"ax, ",[507,58299,21012],{"class":2155},[507,58301,573],{"class":572},[507,58303,2216],{"class":513},[507,58305,58307],{"class":58306},"sVyAn","'$P_e$'",[507,58309,587],{"class":517},[507,58311,58312,58315,58318,58320,58323],{"class":509,"line":553},[507,58313,58314],{"class":517},"ax.",[507,58316,58317],{"class":576},"set_xlabel",[507,58319,580],{"class":517},[507,58321,58322],{"class":730},"'Drive frequency f (GHz)'",[507,58324,587],{"class":517},[507,58326,58327,58329,58332,58334,58337],{"class":509,"line":559},[507,58328,58314],{"class":517},[507,58330,58331],{"class":576},"set_ylabel",[507,58333,580],{"class":517},[507,58335,58336],{"class":730},"'Pulse duration τ (ns)'",[507,58338,587],{"class":517},[507,58340,58341,58343,58346,58348,58351,58354,58356,58358,58361],{"class":509,"line":566},[507,58342,58314],{"class":517},[507,58344,58345],{"class":576},"set_title",[507,58347,580],{"class":517},[507,58349,58350],{"class":730},"'Rabi excited-state probability $P_e(f,",[507,58352,58353],{"class":572},"\\\\",[507,58355,2819],{"class":730},[507,58357,58353],{"class":572},[507,58359,58360],{"class":730},"tau)$'",[507,58362,587],{"class":517},[507,58364,58365,58368,58371],{"class":509,"line":590},[507,58366,58367],{"class":517},"fig.",[507,58369,58370],{"class":576},"tight_layout",[507,58372,781],{"class":517},[507,58374,58375,58377,58379],{"class":509,"line":610},[507,58376,25376],{"class":517},[507,58378,25613],{"class":576},[507,58380,781],{"class":517},[831,58382],{"alt":58383,"src":58384},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[498,58386,58388],{"className":500,"code":58387,"language":502,"meta":104,"style":104},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n\nimport importlib.util\nimport subprocess\nimport sys\n\n# -------------------------------------------------------------------------\n# Control knobs\n# -------------------------------------------------------------------------\nINSTALL_BLOCH_DEPENDENCIES = True\n\nBLOCH_REQUIRED_PACKAGES = {\n    \"numpy\": \"numpy\",\n    \"matplotlib\": \"matplotlib\",\n    \"scipy\": \"scipy\",\n    \"qutip\": \"qutip\",\n}\n\nBLOCH_F0_GHZ = globals().get(\"F0_GHZ\", 5.000)\nBLOCH_OMEGA_ONRESONANCE_MHZ = globals().get(\n    \"OMEGA_ONRESONANCE_MHZ\",\n    20.0,\n)\n\nBLOCH_RESONANT_DETUNING_MHZ = 0.0\nBLOCH_DETUNED_DETUNING_MHZ = 40.0\n\n# Use None to automatically plot one resonant pi pulse.\nBLOCH_DURATION_NS = None\n\nBLOCH_NUM_TIME_POINTS = 300\nBLOCH_FIGSIZE = (8.8, 7.8)\nBLOCH_DPI = 250\nBLOCH_VIEW = [-60, 25]\n\nBLOCH_SHOW_CONTROL_AXIS_ARROWS = True\nBLOCH_SHOW_ARROW_TIP_LABELS = True\nBLOCH_SHOW_ENDPOINT_MARKERS = True\nBLOCH_SHOW_LEGEND = True\nBLOCH_PRINT_SUMMARY = True\n\nCOLOR_RESONANT_TRAJECTORY = \"tab:blue\"\nCOLOR_DETUNED_TRAJECTORY = \"tab:orange\"\nCOLOR_RESONANT_AXIS = \"tab:green\"\nCOLOR_DETUNED_AXIS = \"tab:red\"\nCOLOR_INITIAL_STATE = \"black\"\n\nTRAJECTORY_LINEWIDTH = 2.4\nCONTROL_AXIS_LINEWIDTH = 2.2\nENDPOINT_MARKER_SIZE = 45\nARROW_LENGTH_RATIO = 0.12\nARROW_LABEL_SCALE = 1.16\n\nLEGEND_LOCATION = \"upper left\"\nLEGEND_BBOX_TO_ANCHOR = (1.02, 1.02)\n\n\ndef package_is_available(package_name: str) -> bool:\n    \"\"\"Return True when a package can be imported.\"\"\"\n    return importlib.util.find_spec(package_name) is not None\n\n\ndef install_missing_packages(\n    required_packages: dict[str, str],\n) -> None:\n    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n    if not INSTALL_BLOCH_DEPENDENCIES:\n        return\n\n    missing_package_names = [\n        pip_name\n        for import_name, pip_name in required_packages.items()\n        if not package_is_available(import_name)\n    ]\n\n    if not missing_package_names:\n        return\n\n    try:\n        subprocess.run(\n            [sys.executable, \"-m\", \"pip\", \"install\", \"-q\", \"uv\"],\n            check=True,\n        )\n        subprocess.run(\n            [\n                sys.executable,\n                \"-m\",\n                \"uv\",\n                \"pip\",\n                \"install\",\n                \"--system\",\n                \"-q\",\n                *missing_package_names,\n            ],\n            check=True,\n        )\n    except (subprocess.CalledProcessError, FileNotFoundError):\n        subprocess.run(\n            [\n                sys.executable,\n                \"-m\",\n                \"pip\",\n                \"install\",\n                \"-q\",\n                *missing_package_names,\n            ],\n            check=True,\n        )\n\n\ninstall_missing_packages(BLOCH_REQUIRED_PACKAGES)\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport qutip as qt\nfrom matplotlib.lines import Line2D\n\ntry:\n    from IPython import get_ipython\n\n    ipython = get_ipython()\n    if ipython is not None:\n        ipython.run_line_magic(\"matplotlib\", \"inline\")\nexcept ImportError:\n    pass\n\nplt.rcParams.update(\n    {\n        \"figure.dpi\": BLOCH_DPI,\n    }\n)\n\n\ndef get_pi_pulse_duration_ns(omega_onresonance_mhz: float) -> float:\n    \"\"\"Return the resonant pi-pulse duration in nanoseconds.\"\"\"\n    return 1.0e3 \u002F (2.0 * omega_onresonance_mhz)\n\n\ndef solve_rwa_bloch_trajectory(\n    detuning_mhz: float,\n    duration_ns: float,\n    num_time_points: int,\n) -> dict[str, np.ndarray | float]:\n    \"\"\"Solve the RWA two-level trajectory and return Bloch observables.\"\"\"\n    omega_rad_s = 2.0 * np.pi * BLOCH_OMEGA_ONRESONANCE_MHZ * 1.0e6\n    delta_rad_s = 2.0 * np.pi * detuning_mhz * 1.0e6\n    omega_generalized_rad_s = np.hypot(omega_rad_s, delta_rad_s)\n\n    time_s = np.linspace(0.0, duration_ns * 1.0e-9, num_time_points)\n\n    ground_state = qt.basis(2, 0)\n    excited_state = qt.basis(2, 1)\n    excited_projector = excited_state * excited_state.dag()\n\n    hamiltonian = 0.5 * (\n        delta_rad_s * qt.sigmaz()\n        + omega_rad_s * qt.sigmax()\n    )\n\n    result = qt.sesolve(\n        hamiltonian,\n        ground_state,\n        time_s,\n    )\n\n    x_expectation = np.real(np.asarray(qt.expect(qt.sigmax(), result.states)))\n    y_expectation = np.real(np.asarray(qt.expect(qt.sigmay(), result.states)))\n    z_expectation = np.real(np.asarray(qt.expect(qt.sigmaz(), result.states)))\n\n    excited_probability = np.real(\n        np.asarray(qt.expect(excited_projector, result.states))\n    )\n\n    bloch_vectors = np.vstack(\n        [\n            x_expectation,\n            y_expectation,\n            z_expectation,\n        ]\n    )\n\n    control_axis = np.array(\n        [\n            omega_rad_s \u002F omega_generalized_rad_s,\n            0.0,\n            delta_rad_s \u002F omega_generalized_rad_s,\n        ]\n    )\n\n    return {\n        \"time_ns\": time_s * 1.0e9,\n        \"vectors\": bloch_vectors,\n        \"excited_probability\": excited_probability,\n        \"control_axis\": control_axis,\n        \"generalized_rabi_mhz\": (\n            omega_generalized_rad_s \u002F (2.0 * np.pi * 1.0e6)\n        ),\n    }\n\n\ndef render_empty_bloch_sphere(\n    bloch_sphere: qt.Bloch,\n    fallback_figure: plt.Figure,\n):\n    \"\"\"Render an empty QuTiP Bloch sphere and return its Matplotlib axis.\"\"\"\n    if hasattr(bloch_sphere, \"render\"):\n        bloch_sphere.render()\n    elif hasattr(bloch_sphere, \"make_sphere\"):\n        bloch_sphere.make_sphere()\n    else:\n        bloch_sphere.show()\n\n    axis = getattr(bloch_sphere, \"axes\", None)\n\n    if axis is None and fallback_figure.axes:\n        axis = fallback_figure.axes[-1]\n\n    if axis is None:\n        raise RuntimeError(\"Could not access the Matplotlib Bloch axis.\")\n\n    return axis\n\n\ndef draw_bloch_trajectory(\n    axis,\n    vectors: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a Bloch-vector trajectory on an existing 3D axis.\"\"\"\n    axis.plot(\n        vectors[0],\n        vectors[1],\n        vectors[2],\n        color=color,\n        linewidth=TRAJECTORY_LINEWIDTH,\n        label=label,\n    )\n\n\ndef draw_endpoint_marker(\n    axis,\n    vector: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a marker at the final point of a Bloch trajectory.\"\"\"\n    axis.scatter(\n        vector[0],\n        vector[1],\n        vector[2],\n        color=color,\n        s=ENDPOINT_MARKER_SIZE,\n        depthshade=True,\n        label=label,\n    )\n\n\ndef draw_control_axis_arrow(\n    axis,\n    vector: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a normalized effective-control-axis arrow.\"\"\"\n    axis.quiver(\n        0.0,\n        0.0,\n        0.0,\n        vector[0],\n        vector[1],\n        vector[2],\n        color=color,\n        linewidth=CONTROL_AXIS_LINEWIDTH,\n        arrow_length_ratio=ARROW_LENGTH_RATIO,\n        normalize=False,\n        label=label,\n    )\n\n\ndef add_arrow_tip_label(\n    axis,\n    vector: np.ndarray,\n    text: str,\n    color: str,\n) -> None:\n    \"\"\"Place a text label slightly beyond the arrow tip.\"\"\"\n    label_position = ARROW_LABEL_SCALE * vector\n\n    axis.text(\n        label_position[0],\n        label_position[1],\n        label_position[2],\n        text,\n        color=color,\n        fontsize=10,\n        ha=\"center\",\n        va=\"center\",\n    )\n\n\ndef add_initial_state_label(axis) -> None:\n    \"\"\"Label the initialized ground-state pole.\"\"\"\n    axis.scatter(\n        0.0,\n        0.0,\n        1.0,\n        color=COLOR_INITIAL_STATE,\n        s=ENDPOINT_MARKER_SIZE,\n        depthshade=True,\n    )\n    axis.text(\n        0.0,\n        0.0,\n        1.16,\n        r\"initial $|g\\rangle$\",\n        color=COLOR_INITIAL_STATE,\n        fontsize=9,\n        ha=\"center\",\n        va=\"center\",\n    )\n\n\nif BLOCH_DURATION_NS is None:\n    bloch_duration_ns = get_pi_pulse_duration_ns(\n        BLOCH_OMEGA_ONRESONANCE_MHZ\n    )\nelse:\n    bloch_duration_ns = BLOCH_DURATION_NS\n\nresonant_trajectory = solve_rwa_bloch_trajectory(\n    detuning_mhz=BLOCH_RESONANT_DETUNING_MHZ,\n    duration_ns=bloch_duration_ns,\n    num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\ndetuned_trajectory = solve_rwa_bloch_trajectory(\n    detuning_mhz=BLOCH_DETUNED_DETUNING_MHZ,\n    duration_ns=bloch_duration_ns,\n    num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\nresonant_drive_ghz = (\n    BLOCH_F0_GHZ\n    + BLOCH_RESONANT_DETUNING_MHZ * 1.0e-3\n)\ndetuned_drive_ghz = (\n    BLOCH_F0_GHZ\n    + BLOCH_DETUNED_DETUNING_MHZ * 1.0e-3\n)\n\nfigure = plt.figure(figsize=BLOCH_FIGSIZE)\n\nbloch = qt.Bloch(fig=figure)\nbloch.view = BLOCH_VIEW\nbloch.zlabel = [r\"$|g\\rangle$\", r\"$|e\\rangle$\"]\nbloch.title = (\n    \"Bloch-sphere Rabi trajectories\\n\"\n    rf\"resonant $\\Delta\u002F2\\pi={BLOCH_RESONANT_DETUNING_MHZ:.1f}$ MHz, \"\n    rf\"detuned $\\Delta\u002F2\\pi={BLOCH_DETUNED_DETUNING_MHZ:.1f}$ MHz\"\n)\n\nbloch_axis = render_empty_bloch_sphere(\n    bloch_sphere=bloch,\n    fallback_figure=figure,\n)\n\ndraw_bloch_trajectory(\n    axis=bloch_axis,\n    vectors=resonant_trajectory[\"vectors\"],\n    color=COLOR_RESONANT_TRAJECTORY,\n    label=\"Resonant state trajectory\",\n)\ndraw_bloch_trajectory(\n    axis=bloch_axis,\n    vectors=detuned_trajectory[\"vectors\"],\n    color=COLOR_DETUNED_TRAJECTORY,\n    label=\"Detuned state trajectory\",\n)\n\nif BLOCH_SHOW_ENDPOINT_MARKERS:\n    draw_endpoint_marker(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"vectors\"][:, -1],\n        color=COLOR_RESONANT_TRAJECTORY,\n        label=\"Resonant final state\",\n    )\n    draw_endpoint_marker(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"vectors\"][:, -1],\n        color=COLOR_DETUNED_TRAJECTORY,\n        label=\"Detuned final state\",\n    )\n    add_initial_state_label(bloch_axis)\n\nif BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n    draw_control_axis_arrow(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"control_axis\"],\n        color=COLOR_RESONANT_AXIS,\n        label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n    )\n    draw_control_axis_arrow(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"control_axis\"],\n        color=COLOR_DETUNED_AXIS,\n        label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n    )\n\nif BLOCH_SHOW_ARROW_TIP_LABELS and BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n    add_arrow_tip_label(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"control_axis\"],\n        text=r\"$\\hat{n}_{\\mathrm{res}}$\",\n        color=COLOR_RESONANT_AXIS,\n    )\n    add_arrow_tip_label(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"control_axis\"],\n        text=r\"$\\hat{n}_{\\mathrm{det}}$\",\n        color=COLOR_DETUNED_AXIS,\n    )\n\nif BLOCH_SHOW_LEGEND:\n    legend_handles = [\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_RESONANT_TRAJECTORY,\n            linewidth=TRAJECTORY_LINEWIDTH,\n            label=\"Resonant state trajectory\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_DETUNED_TRAJECTORY,\n            linewidth=TRAJECTORY_LINEWIDTH,\n            label=\"Detuned state trajectory\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_RESONANT_AXIS,\n            linewidth=CONTROL_AXIS_LINEWIDTH,\n            label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_DETUNED_AXIS,\n            linewidth=CONTROL_AXIS_LINEWIDTH,\n            label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n        ),\n    ]\n\n    bloch_axis.legend(\n        handles=legend_handles,\n        loc=LEGEND_LOCATION,\n        bbox_to_anchor=LEGEND_BBOX_TO_ANCHOR,\n        frameon=True,\n        borderaxespad=0.0,\n    )\n\nplt.show()\n\nif BLOCH_PRINT_SUMMARY:\n    resonant_axis = resonant_trajectory[\"control_axis\"]\n    detuned_axis = detuned_trajectory[\"control_axis\"]\n\n    print(\"Bloch-sphere trajectory summary\")\n    print(f\"Natural transition frequency f0: {BLOCH_F0_GHZ:.6f} GHz\")\n    print(f\"Resonant drive frequency: {resonant_drive_ghz:.6f} GHz\")\n    print(f\"Detuned drive frequency: {detuned_drive_ghz:.6f} GHz\")\n    print(f\"On-resonance Rabi rate: {BLOCH_OMEGA_ONRESONANCE_MHZ:.3f} MHz\")\n    print(f\"Plotted duration: {bloch_duration_ns:.3f} ns\")\n    print(\n        \"Resonant control axis n_res: \"\n        f\"({resonant_axis[0]:.6f}, \"\n        f\"{resonant_axis[1]:.6f}, \"\n        f\"{resonant_axis[2]:.6f})\"\n    )\n    print(\n        \"Detuned control axis n_det: \"\n        f\"({detuned_axis[0]:.6f}, \"\n        f\"{detuned_axis[1]:.6f}, \"\n        f\"{detuned_axis[2]:.6f})\"\n    )\n    print(\n        \"Resonant final excited-state probability: \"\n        f\"{resonant_trajectory['excited_probability'][-1]:.6f}\"\n    )\n    print(\n        \"Detuned final excited-state probability: \"\n        f\"{detuned_trajectory['excited_probability'][-1]:.6f}\"\n    )\n    print(\n        \"Detuned generalized Rabi frequency: \"\n        f\"{detuned_trajectory['generalized_rabi_mhz']:.6f} MHz\"\n    )\n",[504,58389,58390,58395,58400,58404,58411,58418,58424,58428,58433,58438,58442,58451,58455,58464,58476,58488,58500,58512,58516,58520,58546,58561,58568,58575,58579,58583,58593,58603,58607,58612,58622,58626,58635,58655,58664,58684,58688,58697,58706,58715,58724,58733,58737,58747,58757,58767,58777,58787,58791,58801,58811,58821,58831,58841,58845,58855,58873,58877,58881,58905,58910,58929,58933,58937,58946,58962,58970,58975,58986,58991,58995,59004,59009,59025,59036,59041,59045,59054,59058,59062,59069,59078,59108,59119,59124,59132,59137,59142,59149,59156,59163,59170,59177,59184,59192,59197,59207,59211,59219,59227,59231,59235,59241,59247,59253,59259,59265,59269,59279,59283,59287,59291,59302,59306,59316,59326,59338,59350,59355,59362,59376,59381,59394,59410,59430,59438,59444,59449,59460,59466,59478,59484,59489,59494,59499,59522,59528,59548,59553,59558,59568,59580,59592,59604,59624,59630,59654,59677,59693,59698,59725,59730,59754,59776,59797,59802,59816,59831,59849,59854,59859,59874,59880,59886,59892,59897,59902,59934,59961,59987,59992,60006,60021,60026,60031,60045,60050,60056,60062,60068,60073,60078,60083,60097,60102,60113,60121,60131,60136,60141,60146,60153,60169,60178,60187,60196,60205,60228,60234,60239,60244,60249,60259,60268,60277,60282,60288,60304,60315,60330,60340,60347,60356,60361,60383,60388,60405,60422,60427,60440,60454,60459,60467,60472,60477,60487,60495,60503,60514,60525,60534,60540,60550,60560,60569,60578,60589,60601,60612,60617,60622,60627,60637,60644,60652,60663,60674,60683,60689,60699,60709,60718,60727,60736,60748,60760,60769,60774,60779,60784,60794,60801,60808,60819,60830,60839,60845,60855,60863,60870,60877,60886,60895,60904,60913,60924,60936,60948,60957,60962,60967,60972,60982,60989,60996,61008,61019,61028,61034,61050,61055,61064,61074,61083,61092,61098,61107,61119,61132,61144,61149,61154,61159,61178,61184,61193,61200,61207,61215,61226,61237,61248,61253,61262,61269,61276,61284,61305,61316,61328,61339,61350,61355,61360,61365,61379,61391,61397,61402,61410,61420,61425,61437,61448,61458,61469,61474,61479,61491,61502,61511,61522,61527,61532,61542,61548,61562,61567,61577,61582,61594,61599,61604,61627,61632,61655,61666,61705,61715,61727,61747,61765,61770,61775,61787,61797,61807,61812,61817,61825,61835,61850,61861,61873,61878,61885,61894,61908,61919,61931,61936,61941,61951,61959,61969,61990,62001,62013,62018,62025,62034,62053,62064,62076,62081,62090,62095,62105,62113,62122,62136,62147,62173,62178,62185,62194,62207,62218,62241,62246,62251,62265,62273,62282,62295,62317,62328,62333,62340,62349,62362,62383,62394,62399,62404,62414,62424,62432,62442,62451,62463,62475,62487,62492,62499,62508,62517,62528,62539,62550,62555,62562,62571,62580,62591,62602,62623,62628,62635,62644,62653,62664,62675,62696,62701,62706,62711,62721,62732,62744,62756,62768,62780,62785,62790,62799,62804,62814,62829,62844,62849,62861,62886,62911,62936,62961,62987,62994,63000,63025,63046,63068,63073,63080,63086,63108,63129,63150,63155,63162,63168,63196,63201,63208,63214,63241,63246,63253,63259,63282],{"__ignoreMap":104},[507,58391,58392],{"class":509,"line":510},[507,58393,58394],{"class":562},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[507,58396,58397],{"class":509,"line":105},[507,58398,58399],{"class":730},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[507,58401,58402],{"class":509,"line":540},[507,58403,556],{"emptyLinePlaceholder":133},[507,58405,58406,58408],{"class":509,"line":553},[507,58407,514],{"class":513},[507,58409,58410],{"class":517}," importlib.util\n",[507,58412,58413,58415],{"class":509,"line":559},[507,58414,514],{"class":513},[507,58416,58417],{"class":517}," subprocess\n",[507,58419,58420,58422],{"class":509,"line":566},[507,58421,514],{"class":513},[507,58423,57106],{"class":517},[507,58425,58426],{"class":509,"line":590},[507,58427,556],{"emptyLinePlaceholder":133},[507,58429,58430],{"class":509,"line":610},[507,58431,58432],{"class":562},"# -------------------------------------------------------------------------\n",[507,58434,58435],{"class":509,"line":634},[507,58436,58437],{"class":562},"# Control knobs\n",[507,58439,58440],{"class":509,"line":661},[507,58441,58432],{"class":562},[507,58443,58444,58447,58449],{"class":509,"line":678},[507,58445,58446],{"class":583},"INSTALL_BLOCH_DEPENDENCIES",[507,58448,1423],{"class":572},[507,58450,57216],{"class":583},[507,58452,58453],{"class":509,"line":683},[507,58454,556],{"emptyLinePlaceholder":133},[507,58456,58457,58460,58462],{"class":509,"line":697},[507,58458,58459],{"class":583},"BLOCH_REQUIRED_PACKAGES",[507,58461,1423],{"class":572},[507,58463,23723],{"class":517},[507,58465,58466,58469,58471,58474],{"class":509,"line":710},[507,58467,58468],{"class":730},"    \"numpy\"",[507,58470,1403],{"class":517},[507,58472,58473],{"class":730},"\"numpy\"",[507,58475,1409],{"class":517},[507,58477,58478,58481,58483,58486],{"class":509,"line":715},[507,58479,58480],{"class":730},"    \"matplotlib\"",[507,58482,1403],{"class":517},[507,58484,58485],{"class":730},"\"matplotlib\"",[507,58487,1409],{"class":517},[507,58489,58490,58493,58495,58498],{"class":509,"line":721},[507,58491,58492],{"class":730},"    \"scipy\"",[507,58494,1403],{"class":517},[507,58496,58497],{"class":730},"\"scipy\"",[507,58499,1409],{"class":517},[507,58501,58502,58505,58507,58510],{"class":509,"line":736},[507,58503,58504],{"class":730},"    \"qutip\"",[507,58506,1403],{"class":517},[507,58508,58509],{"class":730},"\"qutip\"",[507,58511,1409],{"class":517},[507,58513,58514],{"class":509,"line":748},[507,58515,23875],{"class":517},[507,58517,58518],{"class":509,"line":761},[507,58519,556],{"emptyLinePlaceholder":133},[507,58521,58522,58525,58527,58530,58532,58535,58537,58540,58542,58544],{"class":509,"line":775},[507,58523,58524],{"class":583},"BLOCH_F0_GHZ",[507,58526,1423],{"class":572},[507,58528,58529],{"class":572}," globals",[507,58531,13983],{"class":517},[507,58533,58534],{"class":576},"get",[507,58536,580],{"class":517},[507,58538,58539],{"class":730},"\"F0_GHZ\"",[507,58541,622],{"class":517},[507,58543,43802],{"class":583},[507,58545,587],{"class":517},[507,58547,58548,58551,58553,58555,58557,58559],{"class":509,"line":784},[507,58549,58550],{"class":583},"BLOCH_OMEGA_ONRESONANCE_MHZ",[507,58552,1423],{"class":572},[507,58554,58529],{"class":572},[507,58556,13983],{"class":517},[507,58558,58534],{"class":576},[507,58560,1376],{"class":517},[507,58562,58563,58566],{"class":509,"line":796},[507,58564,58565],{"class":730},"    \"OMEGA_ONRESONANCE_MHZ\"",[507,58567,1409],{"class":517},[507,58569,58570,58573],{"class":509,"line":809},[507,58571,58572],{"class":583},"    20.0",[507,58574,1409],{"class":517},[507,58576,58577],{"class":509,"line":1352},[507,58578,587],{"class":517},[507,58580,58581],{"class":509,"line":1357},[507,58582,556],{"emptyLinePlaceholder":133},[507,58584,58585,58588,58590],{"class":509,"line":1362},[507,58586,58587],{"class":583},"BLOCH_RESONANT_DETUNING_MHZ",[507,58589,1423],{"class":572},[507,58591,58592],{"class":583}," 0.0\n",[507,58594,58595,58598,58600],{"class":509,"line":1367},[507,58596,58597],{"class":583},"BLOCH_DETUNED_DETUNING_MHZ",[507,58599,1423],{"class":572},[507,58601,58602],{"class":583}," 40.0\n",[507,58604,58605],{"class":509,"line":1379},[507,58606,556],{"emptyLinePlaceholder":133},[507,58608,58609],{"class":509,"line":1389},[507,58610,58611],{"class":562},"# Use None to automatically plot one resonant pi pulse.\n",[507,58613,58614,58617,58619],{"class":509,"line":1397},[507,58615,58616],{"class":583},"BLOCH_DURATION_NS",[507,58618,1423],{"class":572},[507,58620,58621],{"class":583}," None\n",[507,58623,58624],{"class":509,"line":1412},[507,58625,556],{"emptyLinePlaceholder":133},[507,58627,58628,58631,58633],{"class":509,"line":1431},[507,58629,58630],{"class":583},"BLOCH_NUM_TIME_POINTS",[507,58632,1423],{"class":572},[507,58634,23621],{"class":583},[507,58636,58637,58640,58642,58645,58648,58650,58653],{"class":509,"line":1449},[507,58638,58639],{"class":583},"BLOCH_FIGSIZE",[507,58641,1423],{"class":572},[507,58643,58644],{"class":517}," (",[507,58646,58647],{"class":583},"8.8",[507,58649,622],{"class":517},[507,58651,58652],{"class":583},"7.8",[507,58654,587],{"class":517},[507,58656,58657,58660,58662],{"class":509,"line":1465},[507,58658,58659],{"class":583},"BLOCH_DPI",[507,58661,1423],{"class":572},[507,58663,57252],{"class":583},[507,58665,58666,58669,58671,58673,58675,58678,58680,58682],{"class":509,"line":1471},[507,58667,58668],{"class":583},"BLOCH_VIEW",[507,58670,1423],{"class":572},[507,58672,8427],{"class":517},[507,58674,2367],{"class":572},[507,58676,58677],{"class":583},"60",[507,58679,622],{"class":517},[507,58681,48588],{"class":583},[507,58683,1794],{"class":517},[507,58685,58686],{"class":509,"line":1477},[507,58687,556],{"emptyLinePlaceholder":133},[507,58689,58690,58693,58695],{"class":509,"line":1482},[507,58691,58692],{"class":583},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[507,58694,1423],{"class":572},[507,58696,57216],{"class":583},[507,58698,58699,58702,58704],{"class":509,"line":1488},[507,58700,58701],{"class":583},"BLOCH_SHOW_ARROW_TIP_LABELS",[507,58703,1423],{"class":572},[507,58705,57216],{"class":583},[507,58707,58708,58711,58713],{"class":509,"line":1494},[507,58709,58710],{"class":583},"BLOCH_SHOW_ENDPOINT_MARKERS",[507,58712,1423],{"class":572},[507,58714,57216],{"class":583},[507,58716,58717,58720,58722],{"class":509,"line":1500},[507,58718,58719],{"class":583},"BLOCH_SHOW_LEGEND",[507,58721,1423],{"class":572},[507,58723,57216],{"class":583},[507,58725,58726,58729,58731],{"class":509,"line":1506},[507,58727,58728],{"class":583},"BLOCH_PRINT_SUMMARY",[507,58730,1423],{"class":572},[507,58732,57216],{"class":583},[507,58734,58735],{"class":509,"line":1512},[507,58736,556],{"emptyLinePlaceholder":133},[507,58738,58739,58742,58744],{"class":509,"line":1518},[507,58740,58741],{"class":583},"COLOR_RESONANT_TRAJECTORY",[507,58743,1423],{"class":572},[507,58745,58746],{"class":730}," \"tab:blue\"\n",[507,58748,58749,58752,58754],{"class":509,"line":1524},[507,58750,58751],{"class":583},"COLOR_DETUNED_TRAJECTORY",[507,58753,1423],{"class":572},[507,58755,58756],{"class":730}," \"tab:orange\"\n",[507,58758,58759,58762,58764],{"class":509,"line":1530},[507,58760,58761],{"class":583},"COLOR_RESONANT_AXIS",[507,58763,1423],{"class":572},[507,58765,58766],{"class":730}," \"tab:green\"\n",[507,58768,58769,58772,58774],{"class":509,"line":1536},[507,58770,58771],{"class":583},"COLOR_DETUNED_AXIS",[507,58773,1423],{"class":572},[507,58775,58776],{"class":730}," \"tab:red\"\n",[507,58778,58779,58782,58784],{"class":509,"line":1542},[507,58780,58781],{"class":583},"COLOR_INITIAL_STATE",[507,58783,1423],{"class":572},[507,58785,58786],{"class":730}," \"black\"\n",[507,58788,58789],{"class":509,"line":1548},[507,58790,556],{"emptyLinePlaceholder":133},[507,58792,58793,58796,58798],{"class":509,"line":1553},[507,58794,58795],{"class":583},"TRAJECTORY_LINEWIDTH",[507,58797,1423],{"class":572},[507,58799,58800],{"class":583}," 2.4\n",[507,58802,58803,58806,58808],{"class":509,"line":1559},[507,58804,58805],{"class":583},"CONTROL_AXIS_LINEWIDTH",[507,58807,1423],{"class":572},[507,58809,58810],{"class":583}," 2.2\n",[507,58812,58813,58816,58818],{"class":509,"line":1565},[507,58814,58815],{"class":583},"ENDPOINT_MARKER_SIZE",[507,58817,1423],{"class":572},[507,58819,58820],{"class":583}," 45\n",[507,58822,58823,58826,58828],{"class":509,"line":1570},[507,58824,58825],{"class":583},"ARROW_LENGTH_RATIO",[507,58827,1423],{"class":572},[507,58829,58830],{"class":583}," 0.12\n",[507,58832,58833,58836,58838],{"class":509,"line":1575},[507,58834,58835],{"class":583},"ARROW_LABEL_SCALE",[507,58837,1423],{"class":572},[507,58839,58840],{"class":583}," 1.16\n",[507,58842,58843],{"class":509,"line":1580},[507,58844,556],{"emptyLinePlaceholder":133},[507,58846,58847,58850,58852],{"class":509,"line":1597},[507,58848,58849],{"class":583},"LEGEND_LOCATION",[507,58851,1423],{"class":572},[507,58853,58854],{"class":730}," \"upper left\"\n",[507,58856,58857,58860,58862,58864,58867,58869,58871],{"class":509,"line":1608},[507,58858,58859],{"class":583},"LEGEND_BBOX_TO_ANCHOR",[507,58861,1423],{"class":572},[507,58863,58644],{"class":517},[507,58865,58866],{"class":583},"1.02",[507,58868,622],{"class":517},[507,58870,58866],{"class":583},[507,58872,587],{"class":517},[507,58874,58875],{"class":509,"line":1624},[507,58876,556],{"emptyLinePlaceholder":133},[507,58878,58879],{"class":509,"line":1657},[507,58880,556],{"emptyLinePlaceholder":133},[507,58882,58883,58885,58888,58890,58893,58895,58898,58901,58903],{"class":509,"line":1663},[507,58884,1370],{"class":513},[507,58886,58887],{"class":576}," package_is_available",[507,58889,580],{"class":517},[507,58891,58892],{"class":1382},"package_name",[507,58894,1403],{"class":517},[507,58896,58897],{"class":572},"str",[507,58899,58900],{"class":517},") -> ",[507,58902,1439],{"class":572},[507,58904,1728],{"class":517},[507,58906,58907],{"class":509,"line":1691},[507,58908,58909],{"class":730},"    \"\"\"Return True when a package can be imported.\"\"\"\n",[507,58911,58912,58914,58917,58920,58923,58925,58927],{"class":509,"line":1714},[507,58913,2504],{"class":513},[507,58915,58916],{"class":517}," importlib.util.",[507,58918,58919],{"class":576},"find_spec",[507,58921,58922],{"class":517},"(package_name) ",[507,58924,37008],{"class":513},[507,58926,21980],{"class":513},[507,58928,58621],{"class":583},[507,58930,58931],{"class":509,"line":1731},[507,58932,556],{"emptyLinePlaceholder":133},[507,58934,58935],{"class":509,"line":1740},[507,58936,556],{"emptyLinePlaceholder":133},[507,58938,58939,58941,58944],{"class":509,"line":1769},[507,58940,1370],{"class":513},[507,58942,58943],{"class":576}," install_missing_packages",[507,58945,1376],{"class":517},[507,58947,58948,58951,58954,58956,58958,58960],{"class":509,"line":1777},[507,58949,58950],{"class":1382},"    required_packages",[507,58952,58953],{"class":517},": dict[",[507,58955,58897],{"class":572},[507,58957,622],{"class":517},[507,58959,58897],{"class":572},[507,58961,58067],{"class":517},[507,58963,58964,58966,58968],{"class":509,"line":1797},[507,58965,58900],{"class":517},[507,58967,56764],{"class":583},[507,58969,1728],{"class":517},[507,58971,58972],{"class":509,"line":1805},[507,58973,58974],{"class":730},"    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n",[507,58976,58977,58979,58981,58984],{"class":509,"line":1812},[507,58978,1717],{"class":513},[507,58980,21980],{"class":513},[507,58982,58983],{"class":583}," INSTALL_BLOCH_DEPENDENCIES",[507,58985,1728],{"class":517},[507,58987,58988],{"class":509,"line":1832},[507,58989,58990],{"class":513},"        return\n",[507,58992,58993],{"class":509,"line":1839},[507,58994,556],{"emptyLinePlaceholder":133},[507,58996,58997,59000,59002],{"class":509,"line":1855},[507,58998,58999],{"class":517},"    missing_package_names ",[507,59001,573],{"class":572},[507,59003,2177],{"class":517},[507,59005,59006],{"class":509,"line":1860},[507,59007,59008],{"class":517},"        pip_name\n",[507,59010,59011,59013,59016,59018,59021,59023],{"class":509,"line":1865},[507,59012,2267],{"class":513},[507,59014,59015],{"class":517}," import_name, pip_name ",[507,59017,1636],{"class":513},[507,59019,59020],{"class":517}," required_packages.",[507,59022,22607],{"class":576},[507,59024,781],{"class":517},[507,59026,59027,59029,59031,59033],{"class":509,"line":1886},[507,59028,1734],{"class":513},[507,59030,21980],{"class":513},[507,59032,58887],{"class":576},[507,59034,59035],{"class":517},"(import_name)\n",[507,59037,59038],{"class":509,"line":1891},[507,59039,59040],{"class":517},"    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z_expectation,\n",[507,60069,60071],{"class":509,"line":60070},180,[507,60072,22627],{"class":517},[507,60074,60076],{"class":509,"line":60075},181,[507,60077,1660],{"class":517},[507,60079,60081],{"class":509,"line":60080},182,[507,60082,556],{"emptyLinePlaceholder":133},[507,60084,60086,60089,60091,60093,60095],{"class":509,"line":60085},183,[507,60087,60088],{"class":517},"    control_axis ",[507,60090,573],{"class":572},[507,60092,1616],{"class":517},[507,60094,1619],{"class":576},[507,60096,1376],{"class":517},[507,60098,60100],{"class":509,"line":60099},184,[507,60101,22581],{"class":517},[507,60103,60105,60108,60110],{"class":509,"line":60104},185,[507,60106,60107],{"class":517},"            omega_rad_s ",[507,60109,645],{"class":572},[507,60111,60112],{"class":517}," omega_generalized_rad_s,\n",[507,60114,60116,60119],{"class":509,"line":60115},186,[507,60117,60118],{"class":583},"            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   bloch_axis.",[507,62718,25558],{"class":576},[507,62720,1376],{"class":517},[507,62722,62724,62727,62729],{"class":509,"line":62723},458,[507,62725,62726],{"class":2155},"        handles",[507,62728,573],{"class":572},[507,62730,62731],{"class":517},"legend_handles,\n",[507,62733,62735,62738,62740,62742],{"class":509,"line":62734},459,[507,62736,62737],{"class":2155},"        loc",[507,62739,573],{"class":572},[507,62741,58849],{"class":583},[507,62743,1409],{"class":517},[507,62745,62747,62750,62752,62754],{"class":509,"line":62746},460,[507,62748,62749],{"class":2155},"        bbox_to_anchor",[507,62751,573],{"class":572},[507,62753,58859],{"class":583},[507,62755,1409],{"class":517},[507,62757,62759,62762,62764,62766],{"class":509,"line":62758},461,[507,62760,62761],{"class":2155},"        frameon",[507,62763,573],{"class":572},[507,62765,13878],{"class":583},[507,62767,1409],{"class":517},[507,62769,62771,62774,62776,62778],{"class":509,"line":62770},462,[507,62772,62773],{"class":2155},"        borderaxespad",[507,62775,573],{"class":572},[507,62777,56714],{"class":583},[507,62779,1409],{"class":517},[507,62781,62783],{"class":509,"line":62782},463,[507,62784,1660],{"class":517},[507,62786,62788],{"class":509,"line":62787},464,[507,62789,556],{"emptyLinePlaceholder":133},[507,62791,62793,62795,62797],{"class":509,"line":62792},465,[507,62794,25376],{"class":517},[507,62796,25613],{"class":576},[507,62798,781],{"class":517},[507,62800,62802],{"class":509,"line":62801},466,[507,62803,556],{"emptyLinePlaceholder":133},[507,62805,62807,62809,62812],{"class":509,"line":62806},467,[507,62808,1645],{"class":513},[507,62810,62811],{"class":583}," BLOCH_PRINT_SUMMARY",[507,62813,1728],{"class":517},[507,62815,62817,62820,62822,62825,62827],{"class":509,"line":62816},468,[507,62818,62819],{"class":517},"    resonant_axis ",[507,62821,573],{"class":572},[507,62823,62824],{"class":517}," resonant_trajectory[",[507,62826,62133],{"class":730},[507,62828,1794],{"class":517},[507,62830,62832,62835,62837,62840,62842],{"class":509,"line":62831},469,[507,62833,62834],{"class":517},"    detuned_axis ",[507,62836,573],{"class":572},[507,62838,62839],{"class":517}," detuned_trajectory[",[507,62841,62133],{"class":730},[507,62843,1794],{"class":517},[507,62845,62847],{"class":509,"line":62846},470,[507,62848,556],{"emptyLinePlaceholder":133},[507,62850,62852,62854,62856,62859],{"class":509,"line":62851},471,[507,62853,2060],{"class":572},[507,62855,580],{"class":517},[507,62857,62858],{"class":730},"\"Bloch-sphere trajectory summary\"",[507,62860,587],{"class":517},[507,62862,62864,62866,62868,62870,62873,62876,62879,62881,62884],{"class":509,"line":62863},472,[507,62865,2060],{"class":572},[507,62867,580],{"class":517},[507,62869,22278],{"class":513},[507,62871,62872],{"class":730},"\"Natural transition frequency f0: ",[507,62874,62875],{"class":583},"{BLOCH_F0_GHZ",[507,62877,62878],{"class":513},":.6f",[507,62880,2872],{"class":583},[507,62882,62883],{"class":730}," GHz\"",[507,62885,587],{"class":517},[507,62887,62889,62891,62893,62895,62898,62900,62903,62905,62907,62909],{"class":509,"line":62888},473,[507,62890,2060],{"class":572},[507,62892,580],{"class":517},[507,62894,22278],{"class":513},[507,62896,62897],{"class":730},"\"Resonant drive frequency: ",[507,62899,2810],{"class":583},[507,62901,62902],{"class":517},"resonant_drive_ghz",[507,62904,62878],{"class":513},[507,62906,2872],{"class":583},[507,62908,62883],{"class":730},[507,62910,587],{"class":517},[507,62912,62914,62916,62918,62920,62923,62925,62928,62930,62932,62934],{"class":509,"line":62913},474,[507,62915,2060],{"class":572},[507,62917,580],{"class":517},[507,62919,22278],{"class":513},[507,62921,62922],{"class":730},"\"Detuned drive frequency: ",[507,62924,2810],{"class":583},[507,62926,62927],{"class":517},"detuned_drive_ghz",[507,62929,62878],{"class":513},[507,62931,2872],{"class":583},[507,62933,62883],{"class":730},[507,62935,587],{"class":517},[507,62937,62939,62941,62943,62945,62948,62951,62954,62956,62959],{"class":509,"line":62938},475,[507,62940,2060],{"class":572},[507,62942,580],{"class":517},[507,62944,22278],{"class":513},[507,62946,62947],{"class":730},"\"On-resonance Rabi rate: ",[507,62949,62950],{"class":583},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[507,62952,62953],{"class":513},":.3f",[507,62955,2872],{"class":583},[507,62957,62958],{"class":730}," MHz\"",[507,62960,587],{"class":517},[507,62962,62964,62966,62968,62970,62973,62975,62978,62980,62982,62985],{"class":509,"line":62963},476,[507,62965,2060],{"class":572},[507,62967,580],{"class":517},[507,62969,22278],{"class":513},[507,62971,62972],{"class":730},"\"Plotted duration: ",[507,62974,2810],{"class":583},[507,62976,62977],{"class":517},"bloch_duration_ns",[507,62979,62953],{"class":513},[507,62981,2872],{"class":583},[507,62983,62984],{"class":730}," ns\"",[507,62986,587],{"class":517},[507,62988,62990,62992],{"class":509,"line":62989},477,[507,62991,2060],{"class":572},[507,62993,1376],{"class":517},[507,62995,62997],{"class":509,"line":62996},478,[507,62998,62999],{"class":730},"        \"Resonant control axis n_res: \"\n",[507,63001,63003,63006,63009,63011,63014,63016,63018,63020,63022],{"class":509,"line":63002},479,[507,63004,63005],{"class":513},"        f",[507,63007,63008],{"class":730},"\"(",[507,63010,2810],{"class":583},[507,63012,63013],{"class":517},"resonant_axis[",[507,63015,601],{"class":583},[507,63017,12273],{"class":517},[507,63019,62878],{"class":513},[507,63021,2872],{"class":583},[507,63023,63024],{"class":730},", \"\n",[507,63026,63028,63030,63032,63034,63036,63038,63040,63042,63044],{"class":509,"line":63027},480,[507,63029,63005],{"class":513},[507,63031,22281],{"class":730},[507,63033,2810],{"class":583},[507,63035,63013],{"class":517},[507,63037,625],{"class":583},[507,63039,12273],{"class":517},[507,63041,62878],{"class":513},[507,63043,2872],{"class":583},[507,63045,63024],{"class":730},[507,63047,63049,63051,63053,63055,63057,63059,63061,63063,63065],{"class":509,"line":63048},481,[507,63050,63005],{"class":513},[507,63052,22281],{"class":730},[507,63054,2810],{"class":583},[507,63056,63013],{"class":517},[507,63058,584],{"class":583},[507,63060,12273],{"class":517},[507,63062,62878],{"class":513},[507,63064,2872],{"class":583},[507,63066,63067],{"class":730},")\"\n",[507,63069,63071],{"class":509,"line":63070},482,[507,63072,1660],{"class":517},[507,63074,63076,63078],{"class":509,"line":63075},483,[507,63077,2060],{"class":572},[507,63079,1376],{"class":517},[507,63081,63083],{"class":509,"line":63082},484,[507,63084,63085],{"class":730},"        \"Detuned control axis n_det: \"\n",[507,63087,63089,63091,63093,63095,63098,63100,63102,63104,63106],{"class":509,"line":63088},485,[507,63090,63005],{"class":513},[507,63092,63008],{"class":730},[507,63094,2810],{"class":583},[507,63096,63097],{"class":517},"detuned_axis[",[507,63099,601],{"class":583},[507,63101,12273],{"class":517},[507,63103,62878],{"class":513},[507,63105,2872],{"class":583},[507,63107,63024],{"class":730},[507,63109,63111,63113,63115,63117,63119,63121,63123,63125,63127],{"class":509,"line":63110},486,[507,63112,63005],{"class":513},[507,63114,22281],{"class":730},[507,63116,2810],{"class":583},[507,63118,63097],{"class":517},[507,63120,625],{"class":583},[507,63122,12273],{"class":517},[507,63124,62878],{"class":513},[507,63126,2872],{"class":583},[507,63128,63024],{"class":730},[507,63130,63132,63134,63136,63138,63140,63142,63144,63146,63148],{"class":509,"line":63131},487,[507,63133,63005],{"class":513},[507,63135,22281],{"class":730},[507,63137,2810],{"class":583},[507,63139,63097],{"class":517},[507,63141,584],{"class":583},[507,63143,12273],{"class":517},[507,63145,62878],{"class":513},[507,63147,2872],{"class":583},[507,63149,63067],{"class":730},[507,63151,63153],{"class":509,"line":63152},488,[507,63154,1660],{"class":517},[507,63156,63158,63160],{"class":509,"line":63157},489,[507,63159,2060],{"class":572},[507,63161,1376],{"class":517},[507,63163,63165],{"class":509,"line":63164},490,[507,63166,63167],{"class":730},"        \"Resonant final excited-state probability: \"\n",[507,63169,63171,63173,63175,63177,63179,63182,63184,63186,63188,63190,63192,63194],{"class":509,"line":63170},491,[507,63172,63005],{"class":513},[507,63174,22281],{"class":730},[507,63176,2810],{"class":583},[507,63178,61844],{"class":517},[507,63180,63181],{"class":730},"'excited_probability'",[507,63183,1755],{"class":517},[507,63185,2367],{"class":572},[507,63187,625],{"class":583},[507,63189,12273],{"class":517},[507,63191,62878],{"class":513},[507,63193,2872],{"class":583},[507,63195,61726],{"class":730},[507,63197,63199],{"class":509,"line":63198},492,[507,63200,1660],{"class":517},[507,63202,63204,63206],{"class":509,"line":63203},493,[507,63205,2060],{"class":572},[507,63207,1376],{"class":517},[507,63209,63211],{"class":509,"line":63210},494,[507,63212,63213],{"class":730},"        \"Detuned final excited-state probability: \"\n",[507,63215,63217,63219,63221,63223,63225,63227,63229,63231,63233,63235,63237,63239],{"class":509,"line":63216},495,[507,63218,63005],{"class":513},[507,63220,22281],{"class":730},[507,63222,2810],{"class":583},[507,63224,61903],{"class":517},[507,63226,63181],{"class":730},[507,63228,1755],{"class":517},[507,63230,2367],{"class":572},[507,63232,625],{"class":583},[507,63234,12273],{"class":517},[507,63236,62878],{"class":513},[507,63238,2872],{"class":583},[507,63240,61726],{"class":730},[507,63242,63244],{"class":509,"line":63243},496,[507,63245,1660],{"class":517},[507,63247,63249,63251],{"class":509,"line":63248},497,[507,63250,2060],{"class":572},[507,63252,1376],{"class":517},[507,63254,63256],{"class":509,"line":63255},498,[507,63257,63258],{"class":730},"        \"Detuned generalized Rabi frequency: \"\n",[507,63260,63262,63264,63266,63268,63270,63273,63275,63277,63279],{"class":509,"line":63261},499,[507,63263,63005],{"class":513},[507,63265,22281],{"class":730},[507,63267,2810],{"class":583},[507,63269,61903],{"class":517},[507,63271,63272],{"class":730},"'generalized_rabi_mhz'",[507,63274,12273],{"class":517},[507,63276,62878],{"class":513},[507,63278,2872],{"class":583},[507,63280,63281],{"class":730}," MHz\"\n",[507,63283,63285],{"class":509,"line":63284},500,[507,63286,1660],{"class":517},[831,63288],{"alt":63289,"src":63290},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[498,63292,63295],{"className":63293,"code":63294,"language":7039,"meta":104},[8531],"Bloch-sphere trajectory summary\nNatural transition frequency f0: 5.000000 GHz\nResonant drive frequency: 5.000000 GHz\nDetuned drive frequency: 5.040000 GHz\nOn-resonance Rabi rate: 20.000 MHz\nPlotted duration: 25.000 ns\nResonant control axis n_res: (1.000000, 0.000000, 0.000000)\nDetuned control axis n_det: (0.447214, 0.000000, 0.894427)\nResonant final excited-state probability: 1.000000\nDetuned final excited-state probability: 0.026264\nDetuned generalized Rabi frequency: 44.721360 MHz\n",[504,63296,63294],{"__ignoreMap":104},[498,63298,63300],{"className":500,"code":63299,"language":502,"meta":104,"style":104},"# --- High-resolution 3D surface with colormap & legend  ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Controls for surface quality and appearance\nUPSAMPLE_FACTOR: int = 2           # 2=~4× more faces; try 3 for even smoother\nSURFACE_CMAP: str = 'inferno'      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\nZ_MIN, Z_MAX = 0.0, 1.0            # clamp\u002Fnormalize expected Pe range\n\n# --- Bilinear upsampling (SciPy-free) ---\n# 1) Upsample frequency axis for each fixed τ\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(),\n                      UPSAMPLE_FACTOR * (len(freq_ghz) - 1) + 1)\nPe_tau_f = np.empty((len(dur_ns), len(freq_hi)), dtype=float)\nfor i in range(len(dur_ns)):\n    Pe_tau_f[i, :] = np.interp(freq_hi, freq_ghz, Pe[i, :])\n\n# 2) Upsample τ axis for each fixed frequency\ndur_hi = np.linspace(dur_ns.min(), dur_ns.max(),\n                     UPSAMPLE_FACTOR * (len(dur_ns) - 1) + 1)\nPe_hi = np.empty((len(dur_hi), len(freq_hi)), dtype=float)\nfor j in range(len(freq_hi)):\n    Pe_hi[:, j] = np.interp(dur_hi, dur_ns, Pe_tau_f[:, j])\n\n# Mesh for plotting\nF_hi, T_hi = np.meshgrid(freq_hi, dur_hi, indexing='xy')\n\n# --- Plot ---\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection='3d')\n\nsurf = ax.plot_surface(\n    F_hi, T_hi, Pe_hi,\n    rstride=1, cstride=1,             # use every row\u002Fcolumn (after upsampling)\n    linewidth=0, antialiased=True,\n    cmap=SURFACE_CMAP,\n    vmin=Z_MIN, vmax=Z_MAX,\n)\n\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$\\tau$ (ns)')\nax.set_zlabel(r'$P_e$')\nax.set_title('Rabi $P_e$ surface')\n\n# Colorbar (legend for the colormap)\ncb = fig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08)\ncb.set_label(r'$P_e$')\n\nfig.tight_layout()\nplt.show()\n",[504,63301,63302,63307,63319,63329,63339,63343,63348,63364,63381,63402,63406,63411,63416,63440,63466,63498,63515,63530,63534,63539,63562,63588,63616,63633,63647,63651,63656,63678,63682,63687,63725,63754,63758,63772,63777,63801,63821,63832,63852,63856,63860,63873,63903,63918,63931,63935,63940,63979,63995,63999,64007],{"__ignoreMap":104},[507,63303,63304],{"class":509,"line":510},[507,63305,63306],{"class":562},"# --- High-resolution 3D surface with colormap & legend  ---\n",[507,63308,63309,63311,63314,63316],{"class":509,"line":105},[507,63310,529],{"class":513},[507,63312,63313],{"class":517}," mpl_toolkits.mplot3d ",[507,63315,514],{"class":513},[507,63317,63318],{"class":517}," Axes3D\n",[507,63320,63321,63323,63325,63327],{"class":509,"line":540},[507,63322,514],{"class":513},[507,63324,518],{"class":517},[507,63326,521],{"class":513},[507,63328,524],{"class":517},[507,63330,63331,63333,63335,63337],{"class":509,"line":553},[507,63332,514],{"class":513},[507,63334,57163],{"class":517},[507,63336,521],{"class":513},[507,63338,1159],{"class":517},[507,63340,63341],{"class":509,"line":559},[507,63342,556],{"emptyLinePlaceholder":133},[507,63344,63345],{"class":509,"line":566},[507,63346,63347],{"class":562},"# Controls for surface quality and appearance\n",[507,63349,63350,63353,63355,63357,63359,63361],{"class":509,"line":590},[507,63351,63352],{"class":583},"UPSAMPLE_FACTOR",[507,63354,1403],{"class":517},[507,63356,1420],{"class":572},[507,63358,1423],{"class":572},[507,63360,2316],{"class":583},[507,63362,63363],{"class":562},"           # 2=~4× more faces; try 3 for even smoother\n",[507,63365,63366,63369,63371,63373,63375,63378],{"class":509,"line":610},[507,63367,63368],{"class":583},"SURFACE_CMAP",[507,63370,1403],{"class":517},[507,63372,58897],{"class":572},[507,63374,1423],{"class":572},[507,63376,63377],{"class":730}," 'inferno'",[507,63379,63380],{"class":562},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[507,63382,63383,63386,63388,63391,63393,63395,63397,63399],{"class":509,"line":634},[507,63384,63385],{"class":583},"Z_MIN",[507,63387,622],{"class":517},[507,63389,63390],{"class":583},"Z_MAX",[507,63392,1423],{"class":572},[507,63394,57367],{"class":583},[507,63396,622],{"class":517},[507,63398,57927],{"class":583},[507,63400,63401],{"class":562},"            # clamp\u002Fnormalize expected Pe range\n",[507,63403,63404],{"class":509,"line":661},[507,63405,556],{"emptyLinePlaceholder":133},[507,63407,63408],{"class":509,"line":678},[507,63409,63410],{"class":562},"# --- Bilinear upsampling (SciPy-free) ---\n",[507,63412,63413],{"class":509,"line":683},[507,63414,63415],{"class":562},"# 1) Upsample frequency axis for each fixed τ\n",[507,63417,63418,63421,63423,63425,63427,63430,63432,63435,63437],{"class":509,"line":697},[507,63419,63420],{"class":517},"freq_hi ",[507,63422,573],{"class":572},[507,63424,1616],{"class":517},[507,63426,57946],{"class":576},[507,63428,63429],{"class":517},"(freq_ghz.",[507,63431,25230],{"class":576},[507,63433,63434],{"class":517},"(), freq_ghz.",[507,63436,36712],{"class":576},[507,63438,63439],{"class":517},"(),\n",[507,63441,63442,63445,63447,63449,63451,63454,63456,63458,63460,63462,63464],{"class":509,"line":710},[507,63443,63444],{"class":583},"                      UPSAMPLE_FACTOR",[507,63446,8229],{"class":572},[507,63448,58644],{"class":517},[507,63450,1763],{"class":572},[507,63452,63453],{"class":517},"(freq_ghz) ",[507,63455,2367],{"class":572},[507,63457,1426],{"class":583},[507,63459,655],{"class":517},[507,63461,2107],{"class":572},[507,63463,1426],{"class":583},[507,63465,587],{"class":517},[507,63467,63468,63471,63473,63475,63478,63481,63483,63486,63488,63491,63493,63496],{"class":509,"line":715},[507,63469,63470],{"class":517},"Pe_tau_f ",[507,63472,573],{"class":572},[507,63474,1616],{"class":517},[507,63476,63477],{"class":576},"empty",[507,63479,63480],{"class":517},"((",[507,63482,1763],{"class":572},[507,63484,63485],{"class":517},"(dur_ns), ",[507,63487,1763],{"class":572},[507,63489,63490],{"class":517},"(freq_hi)), ",[507,63492,2156],{"class":2155},[507,63494,63495],{"class":572},"=float",[507,63497,587],{"class":517},[507,63499,63500,63502,63504,63506,63508,63510,63512],{"class":509,"line":721},[507,63501,1630],{"class":513},[507,63503,8246],{"class":517},[507,63505,1636],{"class":513},[507,63507,8221],{"class":572},[507,63509,580],{"class":517},[507,63511,1763],{"class":572},[507,63513,63514],{"class":517},"(dur_ns)):\n",[507,63516,63517,63520,63522,63524,63527],{"class":509,"line":736},[507,63518,63519],{"class":517},"    Pe_tau_f[i, :] ",[507,63521,573],{"class":572},[507,63523,1616],{"class":517},[507,63525,63526],{"class":576},"interp",[507,63528,63529],{"class":517},"(freq_hi, freq_ghz, Pe[i, :])\n",[507,63531,63532],{"class":509,"line":748},[507,63533,556],{"emptyLinePlaceholder":133},[507,63535,63536],{"class":509,"line":761},[507,63537,63538],{"class":562},"# 2) Upsample τ axis for each fixed frequency\n",[507,63540,63541,63544,63546,63548,63550,63553,63555,63558,63560],{"class":509,"line":775},[507,63542,63543],{"class":517},"dur_hi ",[507,63545,573],{"class":572},[507,63547,1616],{"class":517},[507,63549,57946],{"class":576},[507,63551,63552],{"class":517},"(dur_ns.",[507,63554,25230],{"class":576},[507,63556,63557],{"class":517},"(), dur_ns.",[507,63559,36712],{"class":576},[507,63561,63439],{"class":517},[507,63563,63564,63567,63569,63571,63573,63576,63578,63580,63582,63584,63586],{"class":509,"line":784},[507,63565,63566],{"class":583},"                     UPSAMPLE_FACTOR",[507,63568,8229],{"class":572},[507,63570,58644],{"class":517},[507,63572,1763],{"class":572},[507,63574,63575],{"class":517},"(dur_ns) ",[507,63577,2367],{"class":572},[507,63579,1426],{"class":583},[507,63581,655],{"class":517},[507,63583,2107],{"class":572},[507,63585,1426],{"class":583},[507,63587,587],{"class":517},[507,63589,63590,63593,63595,63597,63599,63601,63603,63606,63608,63610,63612,63614],{"class":509,"line":796},[507,63591,63592],{"class":517},"Pe_hi ",[507,63594,573],{"class":572},[507,63596,1616],{"class":517},[507,63598,63477],{"class":576},[507,63600,63480],{"class":517},[507,63602,1763],{"class":572},[507,63604,63605],{"class":517},"(dur_hi), ",[507,63607,1763],{"class":572},[507,63609,63490],{"class":517},[507,63611,2156],{"class":2155},[507,63613,63495],{"class":572},[507,63615,587],{"class":517},[507,63617,63618,63620,63622,63624,63626,63628,63630],{"class":509,"line":809},[507,63619,1630],{"class":513},[507,63621,8680],{"class":517},[507,63623,1636],{"class":513},[507,63625,8221],{"class":572},[507,63627,580],{"class":517},[507,63629,1763],{"class":572},[507,63631,63632],{"class":517},"(freq_hi)):\n",[507,63634,63635,63638,63640,63642,63644],{"class":509,"line":1352},[507,63636,63637],{"class":517},"    Pe_hi[:, j] ",[507,63639,573],{"class":572},[507,63641,1616],{"class":517},[507,63643,63526],{"class":576},[507,63645,63646],{"class":517},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[507,63648,63649],{"class":509,"line":1357},[507,63650,556],{"emptyLinePlaceholder":133},[507,63652,63653],{"class":509,"line":1362},[507,63654,63655],{"class":562},"# Mesh for plotting\n",[507,63657,63658,63661,63663,63665,63667,63670,63672,63674,63676],{"class":509,"line":1367},[507,63659,63660],{"class":517},"F_hi, T_hi ",[507,63662,573],{"class":572},[507,63664,1616],{"class":517},[507,63666,57670],{"class":576},[507,63668,63669],{"class":517},"(freq_hi, dur_hi, ",[507,63671,57676],{"class":2155},[507,63673,573],{"class":572},[507,63675,57681],{"class":730},[507,63677,587],{"class":517},[507,63679,63680],{"class":509,"line":1379},[507,63681,556],{"emptyLinePlaceholder":133},[507,63683,63684],{"class":509,"line":1389},[507,63685,63686],{"class":562},"# --- Plot ---\n",[507,63688,63689,63692,63694,63696,63698,63700,63702,63704,63706,63708,63710,63713,63715,63718,63720,63723],{"class":509,"line":1397},[507,63690,63691],{"class":517},"fig ",[507,63693,573],{"class":572},[507,63695,58225],{"class":517},[507,63697,61616],{"class":576},[507,63699,580],{"class":517},[507,63701,58233],{"class":2155},[507,63703,573],{"class":572},[507,63705,580],{"class":517},[507,63707,35740],{"class":583},[507,63709,622],{"class":517},[507,63711,63712],{"class":583},"6",[507,63714,2213],{"class":517},[507,63716,63717],{"class":2155},"dpi",[507,63719,573],{"class":572},[507,63721,63722],{"class":583},"200",[507,63724,587],{"class":517},[507,63726,63727,63730,63732,63734,63737,63739,63742,63744,63747,63749,63752],{"class":509,"line":1412},[507,63728,63729],{"class":517},"ax ",[507,63731,573],{"class":572},[507,63733,58283],{"class":517},[507,63735,63736],{"class":576},"add_subplot",[507,63738,580],{"class":517},[507,63740,63741],{"class":583},"111",[507,63743,622],{"class":517},[507,63745,63746],{"class":2155},"projection",[507,63748,573],{"class":572},[507,63750,63751],{"class":730},"'3d'",[507,63753,587],{"class":517},[507,63755,63756],{"class":509,"line":1431},[507,63757,556],{"emptyLinePlaceholder":133},[507,63759,63760,63763,63765,63767,63770],{"class":509,"line":1449},[507,63761,63762],{"class":517},"surf ",[507,63764,573],{"class":572},[507,63766,58257],{"class":517},[507,63768,63769],{"class":576},"plot_surface",[507,63771,1376],{"class":517},[507,63773,63774],{"class":509,"line":1465},[507,63775,63776],{"class":517},"    F_hi, T_hi, Pe_hi,\n",[507,63778,63779,63782,63784,63786,63788,63791,63793,63795,63798],{"class":509,"line":1471},[507,63780,63781],{"class":2155},"    rstride",[507,63783,573],{"class":572},[507,63785,625],{"class":583},[507,63787,622],{"class":517},[507,63789,63790],{"class":2155},"cstride",[507,63792,573],{"class":572},[507,63794,625],{"class":583},[507,63796,63797],{"class":517},",             ",[507,63799,63800],{"class":562},"# use every row\u002Fcolumn (after upsampling)\n",[507,63802,63803,63806,63808,63810,63812,63815,63817,63819],{"class":509,"line":1477},[507,63804,63805],{"class":2155},"    linewidth",[507,63807,573],{"class":572},[507,63809,601],{"class":583},[507,63811,622],{"class":517},[507,63813,63814],{"class":2155},"antialiased",[507,63816,573],{"class":572},[507,63818,13878],{"class":583},[507,63820,1409],{"class":517},[507,63822,63823,63826,63828,63830],{"class":509,"line":1482},[507,63824,63825],{"class":2155},"    cmap",[507,63827,573],{"class":572},[507,63829,63368],{"class":583},[507,63831,1409],{"class":517},[507,63833,63834,63837,63839,63841,63843,63846,63848,63850],{"class":509,"line":1488},[507,63835,63836],{"class":2155},"    vmin",[507,63838,573],{"class":572},[507,63840,63385],{"class":583},[507,63842,622],{"class":517},[507,63844,63845],{"class":2155},"vmax",[507,63847,573],{"class":572},[507,63849,63390],{"class":583},[507,63851,1409],{"class":517},[507,63853,63854],{"class":509,"line":1494},[507,63855,587],{"class":517},[507,63857,63858],{"class":509,"line":1500},[507,63859,556],{"emptyLinePlaceholder":133},[507,63861,63862,63864,63866,63868,63871],{"class":509,"line":1506},[507,63863,58314],{"class":517},[507,63865,58317],{"class":576},[507,63867,580],{"class":517},[507,63869,63870],{"class":730},"'f (GHz)'",[507,63872,587],{"class":517},[507,63874,63875,63877,63879,63881,63883,63886,63889,63892,63894,63896,63898,63901],{"class":509,"line":1512},[507,63876,58314],{"class":517},[507,63878,58331],{"class":576},[507,63880,580],{"class":517},[507,63882,2216],{"class":513},[507,63884,63885],{"class":58306},"'$",[507,63887,63888],{"class":572},"\\t",[507,63890,63891],{"class":58306},"au$ ",[507,63893,580],{"class":583},[507,63895,48678],{"class":58306},[507,63897,3649],{"class":583},[507,63899,63900],{"class":58306},"'",[507,63902,587],{"class":517},[507,63904,63905,63907,63910,63912,63914,63916],{"class":509,"line":1518},[507,63906,58314],{"class":517},[507,63908,63909],{"class":576},"set_zlabel",[507,63911,580],{"class":517},[507,63913,2216],{"class":513},[507,63915,58307],{"class":58306},[507,63917,587],{"class":517},[507,63919,63920,63922,63924,63926,63929],{"class":509,"line":1524},[507,63921,58314],{"class":517},[507,63923,58345],{"class":576},[507,63925,580],{"class":517},[507,63927,63928],{"class":730},"'Rabi $P_e$ surface'",[507,63930,587],{"class":517},[507,63932,63933],{"class":509,"line":1530},[507,63934,556],{"emptyLinePlaceholder":133},[507,63936,63937],{"class":509,"line":1536},[507,63938,63939],{"class":562},"# Colorbar (legend for the colormap)\n",[507,63941,63942,63944,63946,63948,63950,63953,63955,63957,63959,63962,63964,63967,63969,63972,63974,63977],{"class":509,"line":1542},[507,63943,58278],{"class":517},[507,63945,573],{"class":572},[507,63947,58283],{"class":517},[507,63949,58286],{"class":576},[507,63951,63952],{"class":517},"(surf, ",[507,63954,58292],{"class":2155},[507,63956,573],{"class":572},[507,63958,58297],{"class":517},[507,63960,63961],{"class":2155},"shrink",[507,63963,573],{"class":572},[507,63965,63966],{"class":583},"0.7",[507,63968,622],{"class":517},[507,63970,63971],{"class":2155},"pad",[507,63973,573],{"class":572},[507,63975,63976],{"class":583},"0.08",[507,63978,587],{"class":517},[507,63980,63981,63984,63987,63989,63991,63993],{"class":509,"line":1548},[507,63982,63983],{"class":517},"cb.",[507,63985,63986],{"class":576},"set_label",[507,63988,580],{"class":517},[507,63990,2216],{"class":513},[507,63992,58307],{"class":58306},[507,63994,587],{"class":517},[507,63996,63997],{"class":509,"line":1553},[507,63998,556],{"emptyLinePlaceholder":133},[507,64000,64001,64003,64005],{"class":509,"line":1559},[507,64002,58367],{"class":517},[507,64004,58370],{"class":576},[507,64006,781],{"class":517},[507,64008,64009,64011,64013],{"class":509,"line":1565},[507,64010,25376],{"class":517},[507,64012,25613],{"class":576},[507,64014,781],{"class":517},[831,64016],{"alt":64017,"src":64018},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[498,64020,64022],{"className":500,"code":64021,"language":502,"meta":104,"style":104},"# @title Cross-sections\n# Slicing controls\nF_CROSS_GHZ: float = 5.000  # drive frequency slice [GHz]\nT_CROSS_NS: float = 100.0   # duration slice [ns]\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef _nearest_index(vec: np.ndarray, value: float, name: str) -> int:\n    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n    vmin, vmax = float(vec.min()), float(vec.max())\n    if value \u003C vmin or value > vmax:\n        print(f\"[Note] {name}={value} is outside sweep; clamping to [{vmin}, {vmax}].\")\n        value = np.clip(value, vmin, vmax)\n    return int(np.argmin(np.abs(vec - value)))\n\n# --- Slice at f = F_CROSS_GHZ ---\nj = _nearest_index(freq_ghz, F_CROSS_GHZ, \"F_CROSS_GHZ\")\nPe_vs_tau = Pe[:, j]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(dur_ns, Pe_vs_tau, color='blue')\nax.set_xlabel(r'$\\tau$ (ns)')\nax.set_ylabel(r'$P_e$')\n# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\nax.set_title(r'$P_e(\\tau)$ at f = ' + f'{freq_ghz[j]:.6f} GHz')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n\n# --- Slice at τ = T_CROSS_NS ---\ni = _nearest_index(dur_ns, T_CROSS_NS, \"T_CROSS_NS\")\nPe_vs_f = Pe[i, :]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(freq_ghz, Pe_vs_f, color='blue')\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$P_e$')\nax.set_title(r'$P_e(f)$ at $\\tau$ = ' + f'{dur_ns[i]:.3f} ns')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n",[504,64023,64024,64029,64034,64049,64065,64069,64079,64089,64093,64129,64134,64159,64181,64230,64244,64268,64272,64277,64298,64308,64312,64339,64358,64384,64398,64403,64449,64466,64474,64482,64486,64491,64512,64522,64526,64552,64569,64581,64595,64641,64657,64665],{"__ignoreMap":104},[507,64025,64026],{"class":509,"line":510},[507,64027,64028],{"class":562},"# @title Cross-sections\n",[507,64030,64031],{"class":509,"line":105},[507,64032,64033],{"class":562},"# Slicing controls\n",[507,64035,64036,64038,64040,64042,64044,64046],{"class":509,"line":540},[507,64037,56789],{"class":583},[507,64039,1403],{"class":517},[507,64041,1406],{"class":572},[507,64043,1423],{"class":572},[507,64045,57287],{"class":583},[507,64047,64048],{"class":562},"  # drive frequency slice [GHz]\n",[507,64050,64051,64053,64055,64057,64059,64062],{"class":509,"line":553},[507,64052,56888],{"class":583},[507,64054,1403],{"class":517},[507,64056,1406],{"class":572},[507,64058,1423],{"class":572},[507,64060,64061],{"class":583}," 100.0",[507,64063,64064],{"class":562},"   # duration slice [ns]\n",[507,64066,64067],{"class":509,"line":559},[507,64068,556],{"emptyLinePlaceholder":133},[507,64070,64071,64073,64075,64077],{"class":509,"line":566},[507,64072,514],{"class":513},[507,64074,518],{"class":517},[507,64076,521],{"class":513},[507,64078,524],{"class":517},[507,64080,64081,64083,64085,64087],{"class":509,"line":590},[507,64082,514],{"class":513},[507,64084,57163],{"class":517},[507,64086,521],{"class":513},[507,64088,1159],{"class":517},[507,64090,64091],{"class":509,"line":610},[507,64092,556],{"emptyLinePlaceholder":133},[507,64094,64095,64097,64100,64102,64105,64108,64111,64113,64115,64117,64119,64121,64123,64125,64127],{"class":509,"line":634},[507,64096,1370],{"class":513},[507,64098,64099],{"class":576}," _nearest_index",[507,64101,580],{"class":517},[507,64103,64104],{"class":1382},"vec",[507,64106,64107],{"class":517},": np.ndarray, ",[507,64109,64110],{"class":1382},"value",[507,64112,1403],{"class":517},[507,64114,1406],{"class":572},[507,64116,622],{"class":517},[507,64118,22008],{"class":1382},[507,64120,1403],{"class":517},[507,64122,58897],{"class":572},[507,64124,58900],{"class":517},[507,64126,1420],{"class":572},[507,64128,1728],{"class":517},[507,64130,64131],{"class":509,"line":661},[507,64132,64133],{"class":730},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[507,64135,64136,64139,64141,64143,64146,64148,64151,64153,64155,64157],{"class":509,"line":678},[507,64137,64138],{"class":517},"    vmin, vmax ",[507,64140,573],{"class":572},[507,64142,59620],{"class":572},[507,64144,64145],{"class":517},"(vec.",[507,64147,25230],{"class":576},[507,64149,64150],{"class":517},"()), ",[507,64152,1406],{"class":572},[507,64154,64145],{"class":517},[507,64156,36712],{"class":576},[507,64158,22087],{"class":517},[507,64160,64161,64163,64166,64168,64171,64174,64176,64178],{"class":509,"line":683},[507,64162,1717],{"class":513},[507,64164,64165],{"class":517}," value ",[507,64167,5677],{"class":572},[507,64169,64170],{"class":517}," vmin ",[507,64172,64173],{"class":513},"or",[507,64175,64165],{"class":517},[507,64177,1651],{"class":572},[507,64179,64180],{"class":517}," vmax:\n",[507,64182,64183,64186,64188,64190,64193,64195,64197,64199,64201,64203,64205,64207,64210,64212,64215,64217,64219,64221,64223,64225,64228],{"class":509,"line":697},[507,64184,64185],{"class":572},"        print",[507,64187,580],{"class":517},[507,64189,22278],{"class":513},[507,64191,64192],{"class":730},"\"[Note] ",[507,64194,2810],{"class":583},[507,64196,22008],{"class":517},[507,64198,2872],{"class":583},[507,64200,573],{"class":730},[507,64202,2810],{"class":583},[507,64204,64110],{"class":517},[507,64206,2872],{"class":583},[507,64208,64209],{"class":730}," is outside sweep; clamping to [",[507,64211,2810],{"class":583},[507,64213,64214],{"class":517},"vmin",[507,64216,2872],{"class":583},[507,64218,622],{"class":730},[507,64220,2810],{"class":583},[507,64222,63845],{"class":517},[507,64224,2872],{"class":583},[507,64226,64227],{"class":730},"].\"",[507,64229,587],{"class":517},[507,64231,64232,64235,64237,64239,64241],{"class":509,"line":710},[507,64233,64234],{"class":517},"        value ",[507,64236,573],{"class":572},[507,64238,1616],{"class":517},[507,64240,57917],{"class":576},[507,64242,64243],{"class":517},"(value, vmin, vmax)\n",[507,64245,64246,64248,64250,64252,64255,64257,64260,64263,64265],{"class":509,"line":715},[507,64247,2504],{"class":513},[507,64249,2095],{"class":572},[507,64251,59917],{"class":517},[507,64253,64254],{"class":576},"argmin",[507,64256,59917],{"class":517},[507,64258,64259],{"class":576},"abs",[507,64261,64262],{"class":517},"(vec ",[507,64264,2367],{"class":572},[507,64266,64267],{"class":517}," value)))\n",[507,64269,64270],{"class":509,"line":721},[507,64271,556],{"emptyLinePlaceholder":133},[507,64273,64274],{"class":509,"line":736},[507,64275,64276],{"class":562},"# --- Slice at f = F_CROSS_GHZ ---\n",[507,64278,64279,64282,64284,64286,64289,64291,64293,64296],{"class":509,"line":748},[507,64280,64281],{"class":517},"j ",[507,64283,573],{"class":572},[507,64285,64099],{"class":576},[507,64287,64288],{"class":517},"(freq_ghz, ",[507,64290,56789],{"class":583},[507,64292,622],{"class":517},[507,64294,64295],{"class":730},"\"F_CROSS_GHZ\"",[507,64297,587],{"class":517},[507,64299,64300,64303,64305],{"class":509,"line":761},[507,64301,64302],{"class":517},"Pe_vs_tau ",[507,64304,573],{"class":572},[507,64306,64307],{"class":517}," Pe[:, j]\n",[507,64309,64310],{"class":509,"line":775},[507,64311,556],{"emptyLinePlaceholder":133},[507,64313,64314,64316,64318,64320,64322,64324,64326,64328,64330,64332,64334,64337],{"class":509,"line":784},[507,64315,58220],{"class":517},[507,64317,573],{"class":572},[507,64319,58225],{"class":517},[507,64321,58228],{"class":576},[507,64323,580],{"class":517},[507,64325,58233],{"class":2155},[507,64327,573],{"class":572},[507,64329,580],{"class":517},[507,64331,63712],{"class":583},[507,64333,622],{"class":517},[507,64335,64336],{"class":583},"3.5",[507,64338,22540],{"class":517},[507,64340,64341,64343,64345,64348,64351,64353,64356],{"class":509,"line":796},[507,64342,58314],{"class":517},[507,64344,25379],{"class":576},[507,64346,64347],{"class":517},"(dur_ns, Pe_vs_tau, ",[507,64349,64350],{"class":2155},"color",[507,64352,573],{"class":572},[507,64354,64355],{"class":730},"'blue'",[507,64357,587],{"class":517},[507,64359,64360,64362,64364,64366,64368,64370,64372,64374,64376,64378,64380,64382],{"class":509,"line":809},[507,64361,58314],{"class":517},[507,64363,58317],{"class":576},[507,64365,580],{"class":517},[507,64367,2216],{"class":513},[507,64369,63885],{"class":58306},[507,64371,63888],{"class":572},[507,64373,63891],{"class":58306},[507,64375,580],{"class":583},[507,64377,48678],{"class":58306},[507,64379,3649],{"class":583},[507,64381,63900],{"class":58306},[507,64383,587],{"class":517},[507,64385,64386,64388,64390,64392,64394,64396],{"class":509,"line":1352},[507,64387,58314],{"class":517},[507,64389,58331],{"class":576},[507,64391,580],{"class":517},[507,64393,2216],{"class":513},[507,64395,58307],{"class":58306},[507,64397,587],{"class":517},[507,64399,64400],{"class":509,"line":1357},[507,64401,64402],{"class":562},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[507,64404,64405,64407,64409,64411,64413,64416,64418,64420,64423,64425,64428,64430,64433,64435,64437,64440,64442,64444,64447],{"class":509,"line":1362},[507,64406,58314],{"class":517},[507,64408,58345],{"class":576},[507,64410,580],{"class":517},[507,64412,2216],{"class":513},[507,64414,64415],{"class":58306},"'$P_e",[507,64417,580],{"class":583},[507,64419,63888],{"class":572},[507,64421,64422],{"class":58306},"au",[507,64424,3649],{"class":583},[507,64426,64427],{"class":58306},"$ at f = '",[507,64429,8313],{"class":572},[507,64431,64432],{"class":513}," f",[507,64434,63900],{"class":730},[507,64436,2810],{"class":583},[507,64438,64439],{"class":517},"freq_ghz[j]",[507,64441,62878],{"class":513},[507,64443,2872],{"class":583},[507,64445,64446],{"class":730}," GHz'",[507,64448,587],{"class":517},[507,64450,64451,64453,64456,64458,64460,64462,64464],{"class":509,"line":1367},[507,64452,58314],{"class":517},[507,64454,64455],{"class":576},"set_ylim",[507,64457,580],{"class":517},[507,64459,56714],{"class":583},[507,64461,622],{"class":517},[507,64463,57927],{"class":583},[507,64465,587],{"class":517},[507,64467,64468,64470,64472],{"class":509,"line":1379},[507,64469,58367],{"class":517},[507,64471,58370],{"class":576},[507,64473,781],{"class":517},[507,64475,64476,64478,64480],{"class":509,"line":1389},[507,64477,25376],{"class":517},[507,64479,25613],{"class":576},[507,64481,781],{"class":517},[507,64483,64484],{"class":509,"line":1397},[507,64485,556],{"emptyLinePlaceholder":133},[507,64487,64488],{"class":509,"line":1412},[507,64489,64490],{"class":562},"# --- Slice at τ = T_CROSS_NS ---\n",[507,64492,64493,64496,64498,64500,64503,64505,64507,64510],{"class":509,"line":1431},[507,64494,64495],{"class":517},"i ",[507,64497,573],{"class":572},[507,64499,64099],{"class":576},[507,64501,64502],{"class":517},"(dur_ns, ",[507,64504,56888],{"class":583},[507,64506,622],{"class":517},[507,64508,64509],{"class":730},"\"T_CROSS_NS\"",[507,64511,587],{"class":517},[507,64513,64514,64517,64519],{"class":509,"line":1449},[507,64515,64516],{"class":517},"Pe_vs_f ",[507,64518,573],{"class":572},[507,64520,64521],{"class":517}," Pe[i, :]\n",[507,64523,64524],{"class":509,"line":1465},[507,64525,556],{"emptyLinePlaceholder":133},[507,64527,64528,64530,64532,64534,64536,64538,64540,64542,64544,64546,64548,64550],{"class":509,"line":1471},[507,64529,58220],{"class":517},[507,64531,573],{"class":572},[507,64533,58225],{"class":517},[507,64535,58228],{"class":576},[507,64537,580],{"class":517},[507,64539,58233],{"class":2155},[507,64541,573],{"class":572},[507,64543,580],{"class":517},[507,64545,63712],{"class":583},[507,64547,622],{"class":517},[507,64549,64336],{"class":583},[507,64551,22540],{"class":517},[507,64553,64554,64556,64558,64561,64563,64565,64567],{"class":509,"line":1477},[507,64555,58314],{"class":517},[507,64557,25379],{"class":576},[507,64559,64560],{"class":517},"(freq_ghz, Pe_vs_f, ",[507,64562,64350],{"class":2155},[507,64564,573],{"class":572},[507,64566,64355],{"class":730},[507,64568,587],{"class":517},[507,64570,64571,64573,64575,64577,64579],{"class":509,"line":1482},[507,64572,58314],{"class":517},[507,64574,58317],{"class":576},[507,64576,580],{"class":517},[507,64578,63870],{"class":730},[507,64580,587],{"class":517},[507,64582,64583,64585,64587,64589,64591,64593],{"class":509,"line":1488},[507,64584,58314],{"class":517},[507,64586,58331],{"class":576},[507,64588,580],{"class":517},[507,64590,2216],{"class":513},[507,64592,58307],{"class":58306},[507,64594,587],{"class":517},[507,64596,64597,64599,64601,64603,64605,64607,64609,64611,64613,64616,64618,64621,64623,64625,64627,64629,64632,64634,64636,64639],{"class":509,"line":1494},[507,64598,58314],{"class":517},[507,64600,58345],{"class":576},[507,64602,580],{"class":517},[507,64604,2216],{"class":513},[507,64606,64415],{"class":58306},[507,64608,580],{"class":583},[507,64610,22278],{"class":58306},[507,64612,3649],{"class":583},[507,64614,64615],{"class":58306},"$ at $",[507,64617,63888],{"class":572},[507,64619,64620],{"class":58306},"au$ = '",[507,64622,8313],{"class":572},[507,64624,64432],{"class":513},[507,64626,63900],{"class":730},[507,64628,2810],{"class":583},[507,64630,64631],{"class":517},"dur_ns[i]",[507,64633,62953],{"class":513},[507,64635,2872],{"class":583},[507,64637,64638],{"class":730}," ns'",[507,64640,587],{"class":517},[507,64642,64643,64645,64647,64649,64651,64653,64655],{"class":509,"line":1500},[507,64644,58314],{"class":517},[507,64646,64455],{"class":576},[507,64648,580],{"class":517},[507,64650,56714],{"class":583},[507,64652,622],{"class":517},[507,64654,57927],{"class":583},[507,64656,587],{"class":517},[507,64658,64659,64661,64663],{"class":509,"line":1506},[507,64660,58367],{"class":517},[507,64662,58370],{"class":576},[507,64664,781],{"class":517},[507,64666,64667,64669,64671],{"class":509,"line":1512},[507,64668,25376],{"class":517},[507,64670,25613],{"class":576},[507,64672,781],{"class":517},[831,64674],{"alt":64675,"src":64676},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[831,64678],{"alt":64679,"src":64680},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[498,64682,64684],{"className":500,"code":64683,"language":502,"meta":104,"style":104},"# @title FFT of Rabi oscillations along tau (controls + computation)\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# -------- Control knobs (FFT & display) --------\nFFT_PAD: int = 8                 # zero-padding factor (2, 4, 8, ...)\nWINDOW: str = \"hann\"             # \"hann\", \"boxcar\"\nDETREND_MEAN: bool = True        # subtract mean along τ before FFT\nAMP_MODE: str = \"magnitude\"      # \"magnitude\" | \"power\" | \"db\"\nNORM_MODE: str = \"global\"        # \"global\" | \"per_f\" (column-wise)\nFREQ_UNITS: str = \"MHz\"          # \"MHz\" or \"GHz\" for Fourier axis\n\n# -------- Helpers --------\ndef _make_window(n: int, kind: str) -> np.ndarray:\n    if kind.lower() == \"hann\":\n        return np.hanning(n)\n    return np.ones(n, dtype=float)  # boxcar\n\ndef rfft_spectrogram(\n    Pe: np.ndarray,\n    dur_ns: np.ndarray,\n    pad: int = 8,\n    window: str = \"hann\",\n    detrend_mean: bool = True,\n    amp_mode: str = \"magnitude\",\n    norm_mode: str = \"global\",\n    freq_units: str = \"MHz\",\n) -> tuple[np.ndarray, np.ndarray]:\n    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n\n    Args:\n        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n        pad: zero-padding factor (≥1).\n        window: \"hann\" or \"boxcar\".\n        detrend_mean: subtract column-wise mean before FFT.\n        amp_mode: \"magnitude\", \"power\", or \"db\".\n        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n\n    Returns:\n        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n        S:     2D nonnegative spectrogram (N_w, N_f).\n    \"\"\"\n    Pe = np.asarray(Pe, float)\n    dur_ns = np.asarray(dur_ns, float)\n    assert Pe.shape[0] == dur_ns.size, \"Pe must be shaped (N_tau, N_f).\"\n\n    # Uniform sampling assumption (true for np.linspace)\n    dt = float(dur_ns[1] - dur_ns[0]) * 1e-9               # [s]\n    n_tau = dur_ns.size\n    nfft = int(2 ** int(np.ceil(np.log2(n_tau))) * max(1, pad))\n    w = _make_window(n_tau, window)\n\n    # Prepare output\n    S_list = []\n    X = Pe.copy()\n    if detrend_mean:\n        X = X - X.mean(axis=0, keepdims=True)\n    X *= w[:, None]\n\n    # Real FFT along τ (axis=0)\n    Xf = np.fft.rfft(X, n=nfft, axis=0)                     # shape (N_w, N_f)\n    f_fft_hz = np.fft.rfftfreq(nfft, d=dt)                  # [Hz]\n\n    # Amplitude selection\n    if amp_mode.lower() == \"power\":\n        S = np.abs(Xf) ** 2\n    else:\n        S = np.abs(Xf)\n\n    # Normalization\n    if norm_mode.lower() == \"per_f\":\n        col_max = np.maximum(S.max(axis=0, keepdims=True), 1e-15)\n        S = S \u002F col_max\n    else:\n        S = S \u002F max(S.max(), 1e-15)\n\n    if amp_mode.lower() == \"db\":\n        S = 20.0 * np.log10(np.maximum(S, 1e-12))  # dB, clipped floor\n\n    # Units\n    if freq_units.upper() == \"GHz\":\n        f_fft = f_fft_hz * 1e-9\n    else:\n        f_fft = f_fft_hz * 1e-6\n\n    return f_fft, S\n\n# Compute spectrogram with current grid\nf_fft_axis, S_fft = rfft_spectrogram(\n    Pe, dur_ns,\n    pad=FFT_PAD,\n    window=WINDOW,\n    detrend_mean=DETREND_MEAN,\n    amp_mode=AMP_MODE,\n    norm_mode=NORM_MODE,\n    freq_units=FREQ_UNITS,\n)\nprint(\"FFT grid:\", S_fft.shape, \"| Fourier freq range:\", f_fft_axis[0], \"→\", f_fft_axis[-1], FREQ_UNITS)\n",[504,64685,64686,64691,64701,64711,64715,64720,64736,64752,64768,64784,64800,64816,64820,64825,64851,64869,64882,64903,64907,64916,64923,64930,64945,64960,64975,64990,65005,65020,65025,65030,65034,65038,65043,65048,65053,65058,65063,65068,65073,65078,65082,65086,65091,65096,65100,65117,65134,65154,65158,65163,65196,65206,65249,65261,65265,65270,65279,65294,65301,65338,65352,65356,65361,65396,65421,65425,65430,65448,65466,65472,65485,65489,65494,65512,65552,65566,65572,65594,65598,65615,65646,65650,65655,65674,65688,65694,65707,65711,65718,65722,65727,65738,65743,65753,65763,65773,65783,65793,65803,65807],{"__ignoreMap":104},[507,64687,64688],{"class":509,"line":510},[507,64689,64690],{"class":562},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[507,64692,64693,64695,64697,64699],{"class":509,"line":105},[507,64694,514],{"class":513},[507,64696,518],{"class":517},[507,64698,521],{"class":513},[507,64700,524],{"class":517},[507,64702,64703,64705,64707,64709],{"class":509,"line":540},[507,64704,514],{"class":513},[507,64706,57163],{"class":517},[507,64708,521],{"class":513},[507,64710,1159],{"class":517},[507,64712,64713],{"class":509,"line":553},[507,64714,556],{"emptyLinePlaceholder":133},[507,64716,64717],{"class":509,"line":559},[507,64718,64719],{"class":562},"# -------- Control knobs (FFT & display) --------\n",[507,64721,64722,64724,64726,64728,64730,64733],{"class":509,"line":566},[507,64723,57005],{"class":583},[507,64725,1403],{"class":517},[507,64727,1420],{"class":572},[507,64729,1423],{"class":572},[507,64731,64732],{"class":583}," 8",[507,64734,64735],{"class":562},"                 # zero-padding factor (2, 4, 8, ...)\n",[507,64737,64738,64740,64742,64744,64746,64749],{"class":509,"line":590},[507,64739,57019],{"class":583},[507,64741,1403],{"class":517},[507,64743,58897],{"class":572},[507,64745,1423],{"class":572},[507,64747,64748],{"class":730}," \"hann\"",[507,64750,64751],{"class":562},"             # \"hann\", \"boxcar\"\n",[507,64753,64754,64756,64758,64760,64762,64765],{"class":509,"line":610},[507,64755,57034],{"class":583},[507,64757,1403],{"class":517},[507,64759,1439],{"class":572},[507,64761,1423],{"class":572},[507,64763,64764],{"class":583}," True",[507,64766,64767],{"class":562},"        # subtract mean along τ before FFT\n",[507,64769,64770,64772,64774,64776,64778,64781],{"class":509,"line":634},[507,64771,57048],{"class":583},[507,64773,1403],{"class":517},[507,64775,58897],{"class":572},[507,64777,1423],{"class":572},[507,64779,64780],{"class":730}," \"magnitude\"",[507,64782,64783],{"class":562},"      # \"magnitude\" | \"power\" | \"db\"\n",[507,64785,64786,64788,64790,64792,64794,64797],{"class":509,"line":661},[507,64787,57063],{"class":583},[507,64789,1403],{"class":517},[507,64791,58897],{"class":572},[507,64793,1423],{"class":572},[507,64795,64796],{"class":730}," \"global\"",[507,64798,64799],{"class":562},"        # \"global\" | \"per_f\" (column-wise)\n",[507,64801,64802,64804,64806,64808,64810,64813],{"class":509,"line":678},[507,64803,57078],{"class":583},[507,64805,1403],{"class":517},[507,64807,58897],{"class":572},[507,64809,1423],{"class":572},[507,64811,64812],{"class":730}," \"MHz\"",[507,64814,64815],{"class":562},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[507,64817,64818],{"class":509,"line":683},[507,64819,556],{"emptyLinePlaceholder":133},[507,64821,64822],{"class":509,"line":697},[507,64823,64824],{"class":562},"# -------- Helpers --------\n",[507,64826,64827,64829,64832,64834,64836,64838,64840,64842,64845,64847,64849],{"class":509,"line":710},[507,64828,1370],{"class":513},[507,64830,64831],{"class":576}," _make_window",[507,64833,580],{"class":517},[507,64835,4420],{"class":1382},[507,64837,1403],{"class":517},[507,64839,1420],{"class":572},[507,64841,622],{"class":517},[507,64843,64844],{"class":1382},"kind",[507,64846,1403],{"class":517},[507,64848,58897],{"class":572},[507,64850,57514],{"class":517},[507,64852,64853,64855,64858,64861,64863,64865,64867],{"class":509,"line":715},[507,64854,1717],{"class":513},[507,64856,64857],{"class":517}," kind.",[507,64859,64860],{"class":576},"lower",[507,64862,1677],{"class":517},[507,64864,1723],{"class":572},[507,64866,64748],{"class":730},[507,64868,1728],{"class":517},[507,64870,64871,64874,64876,64879],{"class":509,"line":721},[507,64872,64873],{"class":513},"        return",[507,64875,1616],{"class":517},[507,64877,64878],{"class":576},"hanning",[507,64880,64881],{"class":517},"(n)\n",[507,64883,64884,64886,64888,64891,64894,64896,64898,64900],{"class":509,"line":736},[507,64885,2504],{"class":513},[507,64887,1616],{"class":517},[507,64889,64890],{"class":576},"ones",[507,64892,64893],{"class":517},"(n, ",[507,64895,2156],{"class":2155},[507,64897,63495],{"class":572},[507,64899,22718],{"class":517},[507,64901,64902],{"class":562},"# boxcar\n",[507,64904,64905],{"class":509,"line":748},[507,64906,556],{"emptyLinePlaceholder":133},[507,64908,64909,64911,64914],{"class":509,"line":761},[507,64910,1370],{"class":513},[507,64912,64913],{"class":576}," rfft_spectrogram",[507,64915,1376],{"class":517},[507,64917,64918,64921],{"class":509,"line":775},[507,64919,64920],{"class":1382},"    Pe",[507,64922,1386],{"class":517},[507,64924,64925,64928],{"class":509,"line":784},[507,64926,64927],{"class":1382},"    dur_ns",[507,64929,1386],{"class":517},[507,64931,64932,64935,64937,64939,64941,64943],{"class":509,"line":796},[507,64933,64934],{"class":1382},"    pad",[507,64936,1403],{"class":517},[507,64938,1420],{"class":572},[507,64940,1423],{"class":572},[507,64942,64732],{"class":583},[507,64944,1409],{"class":517},[507,64946,64947,64950,64952,64954,64956,64958],{"class":509,"line":809},[507,64948,64949],{"class":1382},"    window",[507,64951,1403],{"class":517},[507,64953,58897],{"class":572},[507,64955,1423],{"class":572},[507,64957,64748],{"class":730},[507,64959,1409],{"class":517},[507,64961,64962,64965,64967,64969,64971,64973],{"class":509,"line":1352},[507,64963,64964],{"class":1382},"    detrend_mean",[507,64966,1403],{"class":517},[507,64968,1439],{"class":572},[507,64970,1423],{"class":572},[507,64972,64764],{"class":583},[507,64974,1409],{"class":517},[507,64976,64977,64980,64982,64984,64986,64988],{"class":509,"line":1357},[507,64978,64979],{"class":1382},"    amp_mode",[507,64981,1403],{"class":517},[507,64983,58897],{"class":572},[507,64985,1423],{"class":572},[507,64987,64780],{"class":730},[507,64989,1409],{"class":517},[507,64991,64992,64995,64997,64999,65001,65003],{"class":509,"line":1362},[507,64993,64994],{"class":1382},"    norm_mode",[507,64996,1403],{"class":517},[507,64998,58897],{"class":572},[507,65000,1423],{"class":572},[507,65002,64796],{"class":730},[507,65004,1409],{"class":517},[507,65006,65007,65010,65012,65014,65016,65018],{"class":509,"line":1367},[507,65008,65009],{"class":1382},"    freq_units",[507,65011,1403],{"class":517},[507,65013,58897],{"class":572},[507,65015,1423],{"class":572},[507,65017,64812],{"class":730},[507,65019,1409],{"class":517},[507,65021,65022],{"class":509,"line":1379},[507,65023,65024],{"class":517},") -> tuple[np.ndarray, np.ndarray]:\n",[507,65026,65027],{"class":509,"line":1389},[507,65028,65029],{"class":730},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[507,65031,65032],{"class":509,"line":1397},[507,65033,556],{"emptyLinePlaceholder":133},[507,65035,65036],{"class":509,"line":1412},[507,65037,1485],{"class":730},[507,65039,65040],{"class":509,"line":1431},[507,65041,65042],{"class":730},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[507,65044,65045],{"class":509,"line":1449},[507,65046,65047],{"class":730},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[507,65049,65050],{"class":509,"line":1465},[507,65051,65052],{"class":730},"        pad: zero-padding factor (≥1).\n",[507,65054,65055],{"class":509,"line":1471},[507,65056,65057],{"class":730},"        window: \"hann\" or \"boxcar\".\n",[507,65059,65060],{"class":509,"line":1477},[507,65061,65062],{"class":730},"        detrend_mean: subtract column-wise mean before FFT.\n",[507,65064,65065],{"class":509,"line":1482},[507,65066,65067],{"class":730},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[507,65069,65070],{"class":509,"line":1488},[507,65071,65072],{"class":730},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[507,65074,65075],{"class":509,"line":1494},[507,65076,65077],{"class":730},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[507,65079,65080],{"class":509,"line":1500},[507,65081,556],{"emptyLinePlaceholder":133},[507,65083,65084],{"class":509,"line":1506},[507,65085,1556],{"class":730},[507,65087,65088],{"class":509,"line":1512},[507,65089,65090],{"class":730},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[507,65092,65093],{"class":509,"line":1518},[507,65094,65095],{"class":730},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[507,65097,65098],{"class":509,"line":1524},[507,65099,1468],{"class":730},[507,65101,65102,65105,65107,65109,65111,65113,65115],{"class":509,"line":1530},[507,65103,65104],{"class":517},"    Pe ",[507,65106,573],{"class":572},[507,65108,1616],{"class":517},[507,65110,59920],{"class":576},[507,65112,57920],{"class":517},[507,65114,1406],{"class":572},[507,65116,587],{"class":517},[507,65118,65119,65122,65124,65126,65128,65130,65132],{"class":509,"line":1536},[507,65120,65121],{"class":517},"    dur_ns ",[507,65123,573],{"class":572},[507,65125,1616],{"class":517},[507,65127,59920],{"class":576},[507,65129,64502],{"class":517},[507,65131,1406],{"class":572},[507,65133,587],{"class":517},[507,65135,65136,65139,65142,65144,65146,65148,65151],{"class":509,"line":1542},[507,65137,65138],{"class":513},"    assert",[507,65140,65141],{"class":517}," Pe.shape[",[507,65143,601],{"class":583},[507,65145,8206],{"class":517},[507,65147,1723],{"class":572},[507,65149,65150],{"class":517}," dur_ns.size, ",[507,65152,65153],{"class":730},"\"Pe must be shaped (N_tau, N_f).\"\n",[507,65155,65156],{"class":509,"line":1548},[507,65157,556],{"emptyLinePlaceholder":133},[507,65159,65160],{"class":509,"line":1553},[507,65161,65162],{"class":562},"    # Uniform sampling assumption (true for np.linspace)\n",[507,65164,65165,65168,65170,65172,65175,65177,65179,65181,65184,65186,65189,65191,65193],{"class":509,"line":1559},[507,65166,65167],{"class":517},"    dt ",[507,65169,573],{"class":572},[507,65171,59620],{"class":572},[507,65173,65174],{"class":517},"(dur_ns[",[507,65176,625],{"class":583},[507,65178,8206],{"class":517},[507,65180,2367],{"class":572},[507,65182,65183],{"class":517}," dur_ns[",[507,65185,601],{"class":583},[507,65187,65188],{"class":517},"]) ",[507,65190,2391],{"class":572},[507,65192,57888],{"class":583},[507,65194,65195],{"class":562},"               # [s]\n",[507,65197,65198,65201,65203],{"class":509,"line":1565},[507,65199,65200],{"class":517},"    n_tau ",[507,65202,573],{"class":572},[507,65204,65205],{"class":517}," dur_ns.size\n",[507,65207,65208,65211,65213,65215,65217,65219,65222,65224,65226,65229,65231,65234,65237,65239,65242,65244,65246],{"class":509,"line":1570},[507,65209,65210],{"class":517},"    nfft ",[507,65212,573],{"class":572},[507,65214,2095],{"class":572},[507,65216,580],{"class":517},[507,65218,584],{"class":583},[507,65220,65221],{"class":572}," **",[507,65223,2095],{"class":572},[507,65225,59917],{"class":517},[507,65227,65228],{"class":576},"ceil",[507,65230,59917],{"class":517},[507,65232,65233],{"class":576},"log2",[507,65235,65236],{"class":517},"(n_tau))) ",[507,65238,2391],{"class":572},[507,65240,65241],{"class":572}," max",[507,65243,580],{"class":517},[507,65245,625],{"class":583},[507,65247,65248],{"class":517},", pad))\n",[507,65250,65251,65254,65256,65258],{"class":509,"line":1575},[507,65252,65253],{"class":517},"    w ",[507,65255,573],{"class":572},[507,65257,64831],{"class":576},[507,65259,65260],{"class":517},"(n_tau, window)\n",[507,65262,65263],{"class":509,"line":1580},[507,65264,556],{"emptyLinePlaceholder":133},[507,65266,65267],{"class":509,"line":1597},[507,65268,65269],{"class":562},"    # Prepare output\n",[507,65271,65272,65275,65277],{"class":509,"line":1608},[507,65273,65274],{"class":517},"    S_list ",[507,65276,573],{"class":572},[507,65278,1910],{"class":517},[507,65280,65281,65284,65286,65289,65292],{"class":509,"line":1624},[507,65282,65283],{"class":517},"    X ",[507,65285,573],{"class":572},[507,65287,65288],{"class":517}," Pe.",[507,65290,65291],{"class":576},"copy",[507,65293,781],{"class":517},[507,65295,65296,65298],{"class":509,"line":1657},[507,65297,1717],{"class":513},[507,65299,65300],{"class":517}," detrend_mean:\n",[507,65302,65303,65306,65308,65311,65313,65316,65319,65321,65323,65325,65327,65329,65332,65334,65336],{"class":509,"line":1663},[507,65304,65305],{"class":517},"        X ",[507,65307,573],{"class":572},[507,65309,65310],{"class":517}," X ",[507,65312,2367],{"class":572},[507,65314,65315],{"class":517}," X.",[507,65317,65318],{"class":576},"mean",[507,65320,580],{"class":517},[507,65322,61171],{"class":2155},[507,65324,573],{"class":572},[507,65326,601],{"class":583},[507,65328,622],{"class":517},[507,65330,65331],{"class":2155},"keepdims",[507,65333,573],{"class":572},[507,65335,13878],{"class":583},[507,65337,587],{"class":517},[507,65339,65340,65342,65345,65348,65350],{"class":509,"line":1691},[507,65341,65283],{"class":517},[507,65343,65344],{"class":572},"*=",[507,65346,65347],{"class":517}," w[:, ",[507,65349,56764],{"class":583},[507,65351,1794],{"class":517},[507,65353,65354],{"class":509,"line":1714},[507,65355,556],{"emptyLinePlaceholder":133},[507,65357,65358],{"class":509,"line":1731},[507,65359,65360],{"class":562},"    # Real FFT along τ (axis=0)\n",[507,65362,65363,65366,65368,65371,65374,65377,65379,65381,65384,65386,65388,65390,65393],{"class":509,"line":1740},[507,65364,65365],{"class":517},"    Xf ",[507,65367,573],{"class":572},[507,65369,65370],{"class":517}," np.fft.",[507,65372,65373],{"class":576},"rfft",[507,65375,65376],{"class":517},"(X, ",[507,65378,4420],{"class":2155},[507,65380,573],{"class":572},[507,65382,65383],{"class":517},"nfft, ",[507,65385,61171],{"class":2155},[507,65387,573],{"class":572},[507,65389,601],{"class":583},[507,65391,65392],{"class":517},")                     ",[507,65394,65395],{"class":562},"# shape (N_w, N_f)\n",[507,65397,65398,65401,65403,65405,65408,65411,65413,65415,65418],{"class":509,"line":1769},[507,65399,65400],{"class":517},"    f_fft_hz ",[507,65402,573],{"class":572},[507,65404,65370],{"class":517},[507,65406,65407],{"class":576},"rfftfreq",[507,65409,65410],{"class":517},"(nfft, ",[507,65412,4959],{"class":2155},[507,65414,573],{"class":572},[507,65416,65417],{"class":517},"dt)                  ",[507,65419,65420],{"class":562},"# [Hz]\n",[507,65422,65423],{"class":509,"line":1777},[507,65424,556],{"emptyLinePlaceholder":133},[507,65426,65427],{"class":509,"line":1797},[507,65428,65429],{"class":562},"    # Amplitude selection\n",[507,65431,65432,65434,65437,65439,65441,65443,65446],{"class":509,"line":1805},[507,65433,1717],{"class":513},[507,65435,65436],{"class":517}," amp_mode.",[507,65438,64860],{"class":576},[507,65440,1677],{"class":517},[507,65442,1723],{"class":572},[507,65444,65445],{"class":730}," \"power\"",[507,65447,1728],{"class":517},[507,65449,65450,65453,65455,65457,65459,65462,65464],{"class":509,"line":1812},[507,65451,65452],{"class":517},"        S ",[507,65454,573],{"class":572},[507,65456,1616],{"class":517},[507,65458,64259],{"class":576},[507,65460,65461],{"class":517},"(Xf) ",[507,65463,2377],{"class":572},[507,65465,57789],{"class":583},[507,65467,65468,65470],{"class":509,"line":1832},[507,65469,1800],{"class":513},[507,65471,1728],{"class":517},[507,65473,65474,65476,65478,65480,65482],{"class":509,"line":1839},[507,65475,65452],{"class":517},[507,65477,573],{"class":572},[507,65479,1616],{"class":517},[507,65481,64259],{"class":576},[507,65483,65484],{"class":517},"(Xf)\n",[507,65486,65487],{"class":509,"line":1855},[507,65488,556],{"emptyLinePlaceholder":133},[507,65490,65491],{"class":509,"line":1860},[507,65492,65493],{"class":562},"    # Normalization\n",[507,65495,65496,65498,65501,65503,65505,65507,65510],{"class":509,"line":1865},[507,65497,1717],{"class":513},[507,65499,65500],{"class":517}," norm_mode.",[507,65502,64860],{"class":576},[507,65504,1677],{"class":517},[507,65506,1723],{"class":572},[507,65508,65509],{"class":730}," \"per_f\"",[507,65511,1728],{"class":517},[507,65513,65514,65517,65519,65521,65524,65527,65529,65531,65533,65535,65537,65539,65541,65543,65545,65547,65550],{"class":509,"line":1886},[507,65515,65516],{"class":517},"        col_max ",[507,65518,573],{"class":572},[507,65520,1616],{"class":517},[507,65522,65523],{"class":576},"maximum",[507,65525,65526],{"class":517},"(S.",[507,65528,36712],{"class":576},[507,65530,580],{"class":517},[507,65532,61171],{"class":2155},[507,65534,573],{"class":572},[507,65536,601],{"class":583},[507,65538,622],{"class":517},[507,65540,65331],{"class":2155},[507,65542,573],{"class":572},[507,65544,13878],{"class":583},[507,65546,2213],{"class":517},[507,65548,65549],{"class":583},"1e-15",[507,65551,587],{"class":517},[507,65553,65554,65556,65558,65561,65563],{"class":509,"line":1891},[507,65555,65452],{"class":517},[507,65557,573],{"class":572},[507,65559,65560],{"class":517}," S ",[507,65562,645],{"class":572},[507,65564,65565],{"class":517}," col_max\n",[507,65567,65568,65570],{"class":509,"line":1897},[507,65569,1800],{"class":513},[507,65571,1728],{"class":517},[507,65573,65574,65576,65578,65580,65582,65584,65586,65588,65590,65592],{"class":509,"line":1902},[507,65575,65452],{"class":517},[507,65577,573],{"class":572},[507,65579,65560],{"class":517},[507,65581,645],{"class":572},[507,65583,65241],{"class":572},[507,65585,65526],{"class":517},[507,65587,36712],{"class":576},[507,65589,13950],{"class":517},[507,65591,65549],{"class":583},[507,65593,587],{"class":517},[507,65595,65596],{"class":509,"line":1913},[507,65597,556],{"emptyLinePlaceholder":133},[507,65599,65600,65602,65604,65606,65608,65610,65613],{"class":509,"line":1933},[507,65601,1717],{"class":513},[507,65603,65436],{"class":517},[507,65605,64860],{"class":576},[507,65607,1677],{"class":517},[507,65609,1723],{"class":572},[507,65611,65612],{"class":730}," \"db\"",[507,65614,1728],{"class":517},[507,65616,65617,65619,65621,65623,65625,65627,65630,65632,65634,65637,65640,65643],{"class":509,"line":1973},[507,65618,65452],{"class":517},[507,65620,573],{"class":572},[507,65622,57303],{"class":583},[507,65624,8229],{"class":572},[507,65626,1616],{"class":517},[507,65628,65629],{"class":576},"log10",[507,65631,59917],{"class":517},[507,65633,65523],{"class":576},[507,65635,65636],{"class":517},"(S, ",[507,65638,65639],{"class":583},"1e-12",[507,65641,65642],{"class":517},"))  ",[507,65644,65645],{"class":562},"# dB, clipped floor\n",[507,65647,65648],{"class":509,"line":1978},[507,65649,556],{"emptyLinePlaceholder":133},[507,65651,65652],{"class":509,"line":1988},[507,65653,65654],{"class":562},"    # Units\n",[507,65656,65657,65659,65662,65665,65667,65669,65672],{"class":509,"line":2003},[507,65658,1717],{"class":513},[507,65660,65661],{"class":517}," freq_units.",[507,65663,65664],{"class":576},"upper",[507,65666,1677],{"class":517},[507,65668,1723],{"class":572},[507,65670,65671],{"class":730}," \"GHz\"",[507,65673,1728],{"class":517},[507,65675,65676,65679,65681,65684,65686],{"class":509,"line":2036},[507,65677,65678],{"class":517},"        f_fft ",[507,65680,573],{"class":572},[507,65682,65683],{"class":517}," f_fft_hz ",[507,65685,2391],{"class":572},[507,65687,57628],{"class":583},[507,65689,65690,65692],{"class":509,"line":2041},[507,65691,1800],{"class":513},[507,65693,1728],{"class":517},[507,65695,65696,65698,65700,65702,65704],{"class":509,"line":2052},[507,65697,65678],{"class":517},[507,65699,573],{"class":572},[507,65701,65683],{"class":517},[507,65703,2391],{"class":572},[507,65705,65706],{"class":583}," 1e-6\n",[507,65708,65709],{"class":509,"line":2057},[507,65710,556],{"emptyLinePlaceholder":133},[507,65712,65713,65715],{"class":509,"line":2071},[507,65714,2504],{"class":513},[507,65716,65717],{"class":517}," f_fft, S\n",[507,65719,65720],{"class":509,"line":2076},[507,65721,556],{"emptyLinePlaceholder":133},[507,65723,65724],{"class":509,"line":2087},[507,65725,65726],{"class":562},"# Compute spectrogram with current grid\n",[507,65728,65729,65732,65734,65736],{"class":509,"line":2115},[507,65730,65731],{"class":517},"f_fft_axis, S_fft ",[507,65733,573],{"class":572},[507,65735,64913],{"class":576},[507,65737,1376],{"class":517},[507,65739,65740],{"class":509,"line":2134},[507,65741,65742],{"class":517},"    Pe, dur_ns,\n",[507,65744,65745,65747,65749,65751],{"class":509,"line":2139},[507,65746,64934],{"class":2155},[507,65748,573],{"class":572},[507,65750,57005],{"class":583},[507,65752,1409],{"class":517},[507,65754,65755,65757,65759,65761],{"class":509,"line":2164},[507,65756,64949],{"class":2155},[507,65758,573],{"class":572},[507,65760,57019],{"class":583},[507,65762,1409],{"class":517},[507,65764,65765,65767,65769,65771],{"class":509,"line":2169},[507,65766,64964],{"class":2155},[507,65768,573],{"class":572},[507,65770,57034],{"class":583},[507,65772,1409],{"class":517},[507,65774,65775,65777,65779,65781],{"class":509,"line":2180},[507,65776,64979],{"class":2155},[507,65778,573],{"class":572},[507,65780,57048],{"class":583},[507,65782,1409],{"class":517},[507,65784,65785,65787,65789,65791],{"class":509,"line":2224},[507,65786,64994],{"class":2155},[507,65788,573],{"class":572},[507,65790,57063],{"class":583},[507,65792,1409],{"class":517},[507,65794,65795,65797,65799,65801],{"class":509,"line":2233},[507,65796,65009],{"class":2155},[507,65798,573],{"class":572},[507,65800,57078],{"class":583},[507,65802,1409],{"class":517},[507,65804,65805],{"class":509,"line":2238},[507,65806,587],{"class":517},[507,65808,65809,65811,65813,65816,65819,65822,65825,65827,65829,65832,65834,65836,65838,65840,65842],{"class":509,"line":2249},[507,65810,8525],{"class":572},[507,65812,580],{"class":517},[507,65814,65815],{"class":730},"\"FFT grid:\"",[507,65817,65818],{"class":517},", S_fft.shape, ",[507,65820,65821],{"class":730},"\"| Fourier freq range:\"",[507,65823,65824],{"class":517},", f_fft_axis[",[507,65826,601],{"class":583},[507,65828,1760],{"class":517},[507,65830,65831],{"class":730},"\"→\"",[507,65833,65824],{"class":517},[507,65835,2367],{"class":572},[507,65837,625],{"class":583},[507,65839,1760],{"class":517},[507,65841,57078],{"class":583},[507,65843,587],{"class":517},[498,65845,65848],{"className":65846,"code":65847,"language":7039,"meta":104},[8531],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[504,65849,65847],{"__ignoreMap":104},[498,65851,65853],{"className":500,"code":65852,"language":502,"meta":104,"style":104},"# @title Theoretical Rabi ridge\nimport math\ntwo_pi = 2.0 * math.pi\nf0 = F0_GHZ * 1e9\ndelta_hz = two_pi * (freq_ghz*1e9 - f0)   # rad\u002Fs detuning\nomega_onres = two_pi * OMEGA_ONRESONANCE_MHZ * 1e6  # rad\u002Fs\nOmega_R = np.sqrt(omega_onres**2 + delta_hz**2)     # rad\u002Fs\nridge_hz = Omega_R \u002F (2.0 * math.pi)                # Hz\nif FREQ_UNITS.upper() == \"GHz\":\n    ridge_axis = ridge_hz * 1e-9\nelse:\n    ridge_axis = ridge_hz * 1e-6  # MHz\n",[504,65854,65855,65860,65866,65879,65893,65921,65941,65973,65996,66015,66029,66035],{"__ignoreMap":104},[507,65856,65857],{"class":509,"line":510},[507,65858,65859],{"class":562},"# @title Theoretical Rabi ridge\n",[507,65861,65862,65864],{"class":509,"line":105},[507,65863,514],{"class":513},[507,65865,57135],{"class":517},[507,65867,65868,65871,65873,65875,65877],{"class":509,"line":540},[507,65869,65870],{"class":517},"two_pi ",[507,65872,573],{"class":572},[507,65874,57579],{"class":583},[507,65876,8229],{"class":572},[507,65878,57584],{"class":517},[507,65880,65881,65884,65886,65889,65891],{"class":509,"line":553},[507,65882,65883],{"class":517},"f0 ",[507,65885,573],{"class":572},[507,65887,65888],{"class":583}," F0_GHZ",[507,65890,8229],{"class":572},[507,65892,57599],{"class":583},[507,65894,65895,65898,65900,65902,65904,65907,65909,65912,65915,65918],{"class":509,"line":559},[507,65896,65897],{"class":517},"delta_hz ",[507,65899,573],{"class":572},[507,65901,57638],{"class":517},[507,65903,2391],{"class":572},[507,65905,65906],{"class":517}," (freq_ghz",[507,65908,2391],{"class":572},[507,65910,65911],{"class":583},"1e9",[507,65913,65914],{"class":572}," -",[507,65916,65917],{"class":517}," f0)   ",[507,65919,65920],{"class":562},"# rad\u002Fs detuning\n",[507,65922,65923,65926,65928,65930,65932,65935,65937,65939],{"class":509,"line":566},[507,65924,65925],{"class":517},"omega_onres ",[507,65927,573],{"class":572},[507,65929,57638],{"class":517},[507,65931,2391],{"class":572},[507,65933,65934],{"class":583}," OMEGA_ONRESONANCE_MHZ",[507,65936,8229],{"class":572},[507,65938,57648],{"class":583},[507,65940,57651],{"class":562},[507,65942,65943,65946,65948,65950,65952,65955,65957,65959,65961,65964,65966,65968,65971],{"class":509,"line":590},[507,65944,65945],{"class":517},"Omega_R ",[507,65947,573],{"class":572},[507,65949,1616],{"class":517},[507,65951,9819],{"class":576},[507,65953,65954],{"class":517},"(omega_onres",[507,65956,2377],{"class":572},[507,65958,584],{"class":583},[507,65960,8313],{"class":572},[507,65962,65963],{"class":517}," delta_hz",[507,65965,2377],{"class":572},[507,65967,584],{"class":583},[507,65969,65970],{"class":517},")     ",[507,65972,57705],{"class":562},[507,65974,65975,65978,65980,65982,65984,65986,65988,65990,65993],{"class":509,"line":610},[507,65976,65977],{"class":517},"ridge_hz ",[507,65979,573],{"class":572},[507,65981,57804],{"class":517},[507,65983,645],{"class":572},[507,65985,58644],{"class":517},[507,65987,59542],{"class":583},[507,65989,8229],{"class":572},[507,65991,65992],{"class":517}," math.pi)                ",[507,65994,65995],{"class":562},"# Hz\n",[507,65997,65998,66000,66003,66005,66007,66009,66011,66013],{"class":509,"line":634},[507,65999,1645],{"class":513},[507,66001,66002],{"class":583}," FREQ_UNITS",[507,66004,53],{"class":517},[507,66006,65664],{"class":576},[507,66008,1677],{"class":517},[507,66010,1723],{"class":572},[507,66012,65671],{"class":730},[507,66014,1728],{"class":517},[507,66016,66017,66020,66022,66025,66027],{"class":509,"line":661},[507,66018,66019],{"class":517},"    ridge_axis ",[507,66021,573],{"class":572},[507,66023,66024],{"class":517}," ridge_hz ",[507,66026,2391],{"class":572},[507,66028,57628],{"class":583},[507,66030,66031,66033],{"class":509,"line":678},[507,66032,61407],{"class":513},[507,66034,1728],{"class":517},[507,66036,66037,66039,66041,66043,66045,66048],{"class":509,"line":683},[507,66038,66019],{"class":517},[507,66040,573],{"class":572},[507,66042,66024],{"class":517},[507,66044,2391],{"class":572},[507,66046,66047],{"class":583}," 1e-6",[507,66049,66050],{"class":562},"  # MHz\n",[498,66052,66054],{"className":500,"code":66053,"language":502,"meta":104,"style":104},"# @title 2D heatmap\nfig, ax = plt.subplots(figsize=(7.5, 5.5))\nim = ax.pcolormesh(freq_ghz, f_fft_axis, S_fft, shading=\"auto\", cmap=\"inferno\")\ncb = fig.colorbar(im, ax=ax)\ncb.set_label(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\n\nax.plot(freq_ghz, ridge_axis, \"--\", lw=1.5, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_title(\"Fourier transform along pulse length: $|\\\\mathcal{F}_\\\\tau\\\\{P_e\\\\}|$\")\nax.legend(loc=\"upper right\")\nfig.tight_layout()\nplt.show()\n",[504,66055,66056,66061,66089,66122,66142,66185,66189,66235,66248,66269,66307,66325,66333],{"__ignoreMap":104},[507,66057,66058],{"class":509,"line":510},[507,66059,66060],{"class":562},"# @title 2D heatmap\n",[507,66062,66063,66065,66067,66069,66071,66073,66075,66077,66079,66082,66084,66087],{"class":509,"line":105},[507,66064,58220],{"class":517},[507,66066,573],{"class":572},[507,66068,58225],{"class":517},[507,66070,58228],{"class":576},[507,66072,580],{"class":517},[507,66074,58233],{"class":2155},[507,66076,573],{"class":572},[507,66078,580],{"class":517},[507,66080,66081],{"class":583},"7.5",[507,66083,622],{"class":517},[507,66085,66086],{"class":583},"5.5",[507,66088,22540],{"class":517},[507,66090,66091,66094,66096,66098,66100,66103,66105,66107,66110,66112,66115,66117,66120],{"class":509,"line":540},[507,66092,66093],{"class":517},"im ",[507,66095,573],{"class":572},[507,66097,58257],{"class":517},[507,66099,58260],{"class":576},[507,66101,66102],{"class":517},"(freq_ghz, f_fft_axis, S_fft, ",[507,66104,58266],{"class":2155},[507,66106,573],{"class":572},[507,66108,66109],{"class":730},"\"auto\"",[507,66111,622],{"class":517},[507,66113,66114],{"class":2155},"cmap",[507,66116,573],{"class":572},[507,66118,66119],{"class":730},"\"inferno\"",[507,66121,587],{"class":517},[507,66123,66124,66126,66128,66130,66132,66135,66137,66139],{"class":509,"line":553},[507,66125,58278],{"class":517},[507,66127,573],{"class":572},[507,66129,58283],{"class":517},[507,66131,58286],{"class":576},[507,66133,66134],{"class":517},"(im, ",[507,66136,58292],{"class":2155},[507,66138,573],{"class":572},[507,66140,66141],{"class":517},"ax)\n",[507,66143,66144,66146,66148,66150,66153,66155,66157,66160,66163,66166,66168,66170,66173,66175,66178,66180,66183],{"class":509,"line":559},[507,66145,63983],{"class":517},[507,66147,63986],{"class":576},[507,66149,580],{"class":517},[507,66151,66152],{"class":730},"\"FFT amplitude\"",[507,66154,8313],{"class":572},[507,66156,58644],{"class":517},[507,66158,66159],{"class":730},"\" (dB)\"",[507,66161,66162],{"class":513}," if",[507,66164,66165],{"class":583}," AMP_MODE",[507,66167,53],{"class":517},[507,66169,64860],{"class":576},[507,66171,66172],{"class":517},"()",[507,66174,1723],{"class":572},[507,66176,66177],{"class":730},"\"db\"",[507,66179,8480],{"class":513},[507,66181,66182],{"class":730}," \" (arb.)\"",[507,66184,22540],{"class":517},[507,66186,66187],{"class":509,"line":566},[507,66188,556],{"emptyLinePlaceholder":133},[507,66190,66191,66193,66195,66198,66201,66203,66206,66208,66211,66213,66215,66217,66219,66221,66224,66227,66230,66233],{"class":509,"line":590},[507,66192,58314],{"class":517},[507,66194,25379],{"class":576},[507,66196,66197],{"class":517},"(freq_ghz, ridge_axis, ",[507,66199,66200],{"class":730},"\"--\"",[507,66202,622],{"class":517},[507,66204,66205],{"class":2155},"lw",[507,66207,573],{"class":572},[507,66209,66210],{"class":583},"1.5",[507,66212,622],{"class":517},[507,66214,21012],{"class":2155},[507,66216,573],{"class":572},[507,66218,2216],{"class":513},[507,66220,61680],{"class":58306},[507,66222,66223],{"class":572},"\\O",[507,66225,66226],{"class":58306},"mega_R\u002F2",[507,66228,66229],{"class":572},"\\p",[507,66231,66232],{"class":58306},"i$\"",[507,66234,587],{"class":517},[507,66236,66237,66239,66241,66243,66246],{"class":509,"line":610},[507,66238,58314],{"class":517},[507,66240,58317],{"class":576},[507,66242,580],{"class":517},[507,66244,66245],{"class":730},"\"Drive frequency f (GHz)\"",[507,66247,587],{"class":517},[507,66249,66250,66252,66254,66256,66258,66261,66264,66267],{"class":509,"line":634},[507,66251,58314],{"class":517},[507,66253,58331],{"class":576},[507,66255,580],{"class":517},[507,66257,22278],{"class":513},[507,66259,66260],{"class":730},"\"Fourier frequency (",[507,66262,66263],{"class":583},"{FREQ_UNITS}",[507,66265,66266],{"class":730},")\"",[507,66268,587],{"class":517},[507,66270,66271,66273,66275,66277,66280,66282,66284,66287,66290,66292,66295,66297,66300,66302,66305],{"class":509,"line":661},[507,66272,58314],{"class":517},[507,66274,58345],{"class":576},[507,66276,580],{"class":517},[507,66278,66279],{"class":730},"\"Fourier transform along pulse length: $|",[507,66281,58353],{"class":572},[507,66283,2574],{"class":730},[507,66285,66286],{"class":583},"{F}",[507,66288,66289],{"class":730},"_",[507,66291,58353],{"class":572},[507,66293,66294],{"class":730},"tau",[507,66296,58353],{"class":572},[507,66298,66299],{"class":730},"{P_e",[507,66301,58353],{"class":572},[507,66303,66304],{"class":730},"}|$\"",[507,66306,587],{"class":517},[507,66308,66309,66311,66313,66315,66318,66320,66323],{"class":509,"line":678},[507,66310,58314],{"class":517},[507,66312,25558],{"class":576},[507,66314,580],{"class":517},[507,66316,66317],{"class":2155},"loc",[507,66319,573],{"class":572},[507,66321,66322],{"class":730},"\"upper right\"",[507,66324,587],{"class":517},[507,66326,66327,66329,66331],{"class":509,"line":683},[507,66328,58367],{"class":517},[507,66330,58370],{"class":576},[507,66332,781],{"class":517},[507,66334,66335,66337,66339],{"class":509,"line":697},[507,66336,25376],{"class":517},[507,66338,25613],{"class":576},[507,66340,781],{"class":517},[831,66342],{"alt":66343,"src":66344},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[498,66346,66348],{"className":500,"code":66347,"language":502,"meta":104,"style":104},"# --- Adaptive high-res 3D surface ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# ------ Control knobs ------\nUPSAMPLE_F_FREQ: int = 3       # try 2–3; upsample the drive-frequency axis only\nMAX_FACES: int = 300_000       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\nMIN_ROWS: int = 128            # keep at least this many Fourier-frequency rows\nMAX_ROWS: int = 1024           # hard cap for rows to keep plotting fast\n\n# Preserve global color limits so the legend stays consistent\nvmin = float(np.nanmin(S_fft))\nvmax = float(np.nanmax(S_fft))\n\n# Shapes\nn_rows, n_cols = S_fft.shape        # rows=Fourier freq (ν), cols=drive freq (f)\n\n# 1) Upsample along the typically smaller axis (drive frequency) only\nn_cols_up = min(UPSAMPLE_F_FREQ * (n_cols - 1) + 1, 1200)  # sensible upper bound\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(), n_cols_up)\n\nS_up_f = np.empty((n_rows, n_cols_up), dtype=float)\nfor i in range(n_rows):\n    S_up_f[i, :] = np.interp(freq_hi, freq_ghz, S_fft[i, :])\n\n# 2) Determine how many Fourier rows we can afford for MAX_FACES\n# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\nn_rows_cap = int(MAX_FACES \u002F max(n_cols_up - 1, 1)) + 1\nn_rows_plot = int(np.clip(n_rows_cap, MIN_ROWS, min(MAX_ROWS, n_rows)))\n\n# Down-sample ν-axis evenly to n_rows_plot\nrow_idx = np.linspace(0, n_rows - 1, n_rows_plot).astype(int)\nf_fft_plot = f_fft_axis[row_idx]\nS_final = S_up_f[row_idx, :]\n\n# Mesh for plotting\nF2_hi, FFOUR_hi = np.meshgrid(freq_hi, f_fft_plot, indexing=\"xy\")\n\n# 3) Plot (keeps 'inferno')\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection=\"3d\")\nsurf = ax.plot_surface(\n    F2_hi, FFOUR_hi, S_final,\n    rstride=1, cstride=1,\n    linewidth=0, antialiased=True,\n    cmap=\"inferno\",\n    vmin=vmin, vmax=vmax\n)\n\nax.set_xlabel(\"f (GHz)\")\nax.set_ylabel(f\"Fourier freq ({FREQ_UNITS})\")\nax.set_zlabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower() == \"db\" else \" (arb.)\"))\nax.set_title(\"Rabi FFT surface (adaptive resolution)\")\nfig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08, label=\"Amplitude\")\n\n# A clearer default view\nax.view_init(elev=25, azim=-60)\n\nfig.tight_layout()\nplt.show()\n",[504,66349,66350,66355,66365,66375,66385,66389,66394,66411,66428,66445,66462,66466,66471,66488,66504,66508,66513,66526,66530,66535,66573,66594,66598,66618,66631,66645,66649,66654,66659,66694,66723,66727,66732,66766,66776,66786,66790,66794,66820,66824,66829,66863,66888,66900,66910,66928,66946,66956,66972,66976,66980,66993,67012,67048,67061,67100,67104,67109,67136,67140,67148],{"__ignoreMap":104},[507,66351,66352],{"class":509,"line":510},[507,66353,66354],{"class":562},"# --- Adaptive high-res 3D surface ---\n",[507,66356,66357,66359,66361,66363],{"class":509,"line":105},[507,66358,529],{"class":513},[507,66360,63313],{"class":517},[507,66362,514],{"class":513},[507,66364,63318],{"class":517},[507,66366,66367,66369,66371,66373],{"class":509,"line":540},[507,66368,514],{"class":513},[507,66370,518],{"class":517},[507,66372,521],{"class":513},[507,66374,524],{"class":517},[507,66376,66377,66379,66381,66383],{"class":509,"line":553},[507,66378,514],{"class":513},[507,66380,57163],{"class":517},[507,66382,521],{"class":513},[507,66384,1159],{"class":517},[507,66386,66387],{"class":509,"line":559},[507,66388,556],{"emptyLinePlaceholder":133},[507,66390,66391],{"class":509,"line":566},[507,66392,66393],{"class":562},"# ------ Control knobs ------\n",[507,66395,66396,66399,66401,66403,66405,66408],{"class":509,"line":590},[507,66397,66398],{"class":583},"UPSAMPLE_F_FREQ",[507,66400,1403],{"class":517},[507,66402,1420],{"class":572},[507,66404,1423],{"class":572},[507,66406,66407],{"class":583}," 3",[507,66409,66410],{"class":562},"       # try 2–3; upsample the drive-frequency axis only\n",[507,66412,66413,66416,66418,66420,66422,66425],{"class":509,"line":610},[507,66414,66415],{"class":583},"MAX_FACES",[507,66417,1403],{"class":517},[507,66419,1420],{"class":572},[507,66421,1423],{"class":572},[507,66423,66424],{"class":583}," 300_000",[507,66426,66427],{"class":562},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[507,66429,66430,66433,66435,66437,66439,66442],{"class":509,"line":634},[507,66431,66432],{"class":583},"MIN_ROWS",[507,66434,1403],{"class":517},[507,66436,1420],{"class":572},[507,66438,1423],{"class":572},[507,66440,66441],{"class":583}," 128",[507,66443,66444],{"class":562},"            # keep at least this many Fourier-frequency rows\n",[507,66446,66447,66450,66452,66454,66456,66459],{"class":509,"line":661},[507,66448,66449],{"class":583},"MAX_ROWS",[507,66451,1403],{"class":517},[507,66453,1420],{"class":572},[507,66455,1423],{"class":572},[507,66457,66458],{"class":583}," 1024",[507,66460,66461],{"class":562},"           # hard cap for rows to keep plotting fast\n",[507,66463,66464],{"class":509,"line":678},[507,66465,556],{"emptyLinePlaceholder":133},[507,66467,66468],{"class":509,"line":683},[507,66469,66470],{"class":562},"# Preserve global color limits so the legend stays consistent\n",[507,66472,66473,66476,66478,66480,66482,66485],{"class":509,"line":697},[507,66474,66475],{"class":517},"vmin ",[507,66477,573],{"class":572},[507,66479,59620],{"class":572},[507,66481,59917],{"class":517},[507,66483,66484],{"class":576},"nanmin",[507,66486,66487],{"class":517},"(S_fft))\n",[507,66489,66490,66493,66495,66497,66499,66502],{"class":509,"line":710},[507,66491,66492],{"class":517},"vmax ",[507,66494,573],{"class":572},[507,66496,59620],{"class":572},[507,66498,59917],{"class":517},[507,66500,66501],{"class":576},"nanmax",[507,66503,66487],{"class":517},[507,66505,66506],{"class":509,"line":715},[507,66507,556],{"emptyLinePlaceholder":133},[507,66509,66510],{"class":509,"line":721},[507,66511,66512],{"class":562},"# Shapes\n",[507,66514,66515,66518,66520,66523],{"class":509,"line":736},[507,66516,66517],{"class":517},"n_rows, n_cols ",[507,66519,573],{"class":572},[507,66521,66522],{"class":517}," S_fft.shape        ",[507,66524,66525],{"class":562},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[507,66527,66528],{"class":509,"line":748},[507,66529,556],{"emptyLinePlaceholder":133},[507,66531,66532],{"class":509,"line":761},[507,66533,66534],{"class":562},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[507,66536,66537,66540,66542,66544,66546,66548,66550,66553,66555,66557,66559,66561,66563,66565,66568,66570],{"class":509,"line":775},[507,66538,66539],{"class":517},"n_cols_up ",[507,66541,573],{"class":572},[507,66543,2476],{"class":572},[507,66545,580],{"class":517},[507,66547,66398],{"class":583},[507,66549,8229],{"class":572},[507,66551,66552],{"class":517}," (n_cols ",[507,66554,2367],{"class":572},[507,66556,1426],{"class":583},[507,66558,655],{"class":517},[507,66560,2107],{"class":572},[507,66562,1426],{"class":583},[507,66564,622],{"class":517},[507,66566,66567],{"class":583},"1200",[507,66569,22718],{"class":517},[507,66571,66572],{"class":562},"# sensible upper bound\n",[507,66574,66575,66577,66579,66581,66583,66585,66587,66589,66591],{"class":509,"line":784},[507,66576,63420],{"class":517},[507,66578,573],{"class":572},[507,66580,1616],{"class":517},[507,66582,57946],{"class":576},[507,66584,63429],{"class":517},[507,66586,25230],{"class":576},[507,66588,63434],{"class":517},[507,66590,36712],{"class":576},[507,66592,66593],{"class":517},"(), n_cols_up)\n",[507,66595,66596],{"class":509,"line":796},[507,66597,556],{"emptyLinePlaceholder":133},[507,66599,66600,66603,66605,66607,66609,66612,66614,66616],{"class":509,"line":809},[507,66601,66602],{"class":517},"S_up_f ",[507,66604,573],{"class":572},[507,66606,1616],{"class":517},[507,66608,63477],{"class":576},[507,66610,66611],{"class":517},"((n_rows, n_cols_up), ",[507,66613,2156],{"class":2155},[507,66615,63495],{"class":572},[507,66617,587],{"class":517},[507,66619,66620,66622,66624,66626,66628],{"class":509,"line":1352},[507,66621,1630],{"class":513},[507,66623,8246],{"class":517},[507,66625,1636],{"class":513},[507,66627,8221],{"class":572},[507,66629,66630],{"class":517},"(n_rows):\n",[507,66632,66633,66636,66638,66640,66642],{"class":509,"line":1357},[507,66634,66635],{"class":517},"    S_up_f[i, :] ",[507,66637,573],{"class":572},[507,66639,1616],{"class":517},[507,66641,63526],{"class":576},[507,66643,66644],{"class":517},"(freq_hi, freq_ghz, S_fft[i, :])\n",[507,66646,66647],{"class":509,"line":1362},[507,66648,556],{"emptyLinePlaceholder":133},[507,66650,66651],{"class":509,"line":1367},[507,66652,66653],{"class":562},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[507,66655,66656],{"class":509,"line":1379},[507,66657,66658],{"class":562},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[507,66660,66661,66664,66666,66668,66670,66672,66674,66676,66679,66681,66683,66685,66687,66690,66692],{"class":509,"line":1389},[507,66662,66663],{"class":517},"n_rows_cap ",[507,66665,573],{"class":572},[507,66667,2095],{"class":572},[507,66669,580],{"class":517},[507,66671,66415],{"class":583},[507,66673,57880],{"class":572},[507,66675,65241],{"class":572},[507,66677,66678],{"class":517},"(n_cols_up ",[507,66680,2367],{"class":572},[507,66682,1426],{"class":583},[507,66684,622],{"class":517},[507,66686,625],{"class":583},[507,66688,66689],{"class":517},")) ",[507,66691,2107],{"class":572},[507,66693,2084],{"class":583},[507,66695,66696,66699,66701,66703,66705,66707,66710,66712,66714,66716,66718,66720],{"class":509,"line":1397},[507,66697,66698],{"class":517},"n_rows_plot ",[507,66700,573],{"class":572},[507,66702,2095],{"class":572},[507,66704,59917],{"class":517},[507,66706,57917],{"class":576},[507,66708,66709],{"class":517},"(n_rows_cap, ",[507,66711,66432],{"class":583},[507,66713,622],{"class":517},[507,66715,25230],{"class":572},[507,66717,580],{"class":517},[507,66719,66449],{"class":583},[507,66721,66722],{"class":517},", n_rows)))\n",[507,66724,66725],{"class":509,"line":1412},[507,66726,556],{"emptyLinePlaceholder":133},[507,66728,66729],{"class":509,"line":1431},[507,66730,66731],{"class":562},"# Down-sample ν-axis evenly to n_rows_plot\n",[507,66733,66734,66737,66739,66741,66743,66745,66747,66750,66752,66754,66757,66760,66762,66764],{"class":509,"line":1449},[507,66735,66736],{"class":517},"row_idx ",[507,66738,573],{"class":572},[507,66740,1616],{"class":517},[507,66742,57946],{"class":576},[507,66744,580],{"class":517},[507,66746,601],{"class":583},[507,66748,66749],{"class":517},", n_rows ",[507,66751,2367],{"class":572},[507,66753,1426],{"class":583},[507,66755,66756],{"class":517},", n_rows_plot).",[507,66758,66759],{"class":576},"astype",[507,66761,580],{"class":517},[507,66763,1420],{"class":572},[507,66765,587],{"class":517},[507,66767,66768,66771,66773],{"class":509,"line":1465},[507,66769,66770],{"class":517},"f_fft_plot ",[507,66772,573],{"class":572},[507,66774,66775],{"class":517}," f_fft_axis[row_idx]\n",[507,66777,66778,66781,66783],{"class":509,"line":1471},[507,66779,66780],{"class":517},"S_final ",[507,66782,573],{"class":572},[507,66784,66785],{"class":517}," S_up_f[row_idx, :]\n",[507,66787,66788],{"class":509,"line":1477},[507,66789,556],{"emptyLinePlaceholder":133},[507,66791,66792],{"class":509,"line":1482},[507,66793,63655],{"class":562},[507,66795,66796,66799,66802,66804,66806,66808,66811,66813,66815,66818],{"class":509,"line":1488},[507,66797,66798],{"class":517},"F2_hi, ",[507,66800,66801],{"class":583},"FFOUR_hi",[507,66803,1423],{"class":572},[507,66805,1616],{"class":517},[507,66807,57670],{"class":576},[507,66809,66810],{"class":517},"(freq_hi, f_fft_plot, ",[507,66812,57676],{"class":2155},[507,66814,573],{"class":572},[507,66816,66817],{"class":730},"\"xy\"",[507,66819,587],{"class":517},[507,66821,66822],{"class":509,"line":1494},[507,66823,556],{"emptyLinePlaceholder":133},[507,66825,66826],{"class":509,"line":1500},[507,66827,66828],{"class":562},"# 3) Plot (keeps 'inferno')\n",[507,66830,66831,66833,66835,66837,66839,66841,66843,66845,66847,66849,66851,66853,66855,66857,66859,66861],{"class":509,"line":1506},[507,66832,63691],{"class":517},[507,66834,573],{"class":572},[507,66836,58225],{"class":517},[507,66838,61616],{"class":576},[507,66840,580],{"class":517},[507,66842,58233],{"class":2155},[507,66844,573],{"class":572},[507,66846,580],{"class":517},[507,66848,35740],{"class":583},[507,66850,622],{"class":517},[507,66852,63712],{"class":583},[507,66854,2213],{"class":517},[507,66856,63717],{"class":2155},[507,66858,573],{"class":572},[507,66860,63722],{"class":583},[507,66862,587],{"class":517},[507,66864,66865,66867,66869,66871,66873,66875,66877,66879,66881,66883,66886],{"class":509,"line":1512},[507,66866,63729],{"class":517},[507,66868,573],{"class":572},[507,66870,58283],{"class":517},[507,66872,63736],{"class":576},[507,66874,580],{"class":517},[507,66876,63741],{"class":583},[507,66878,622],{"class":517},[507,66880,63746],{"class":2155},[507,66882,573],{"class":572},[507,66884,66885],{"class":730},"\"3d\"",[507,66887,587],{"class":517},[507,66889,66890,66892,66894,66896,66898],{"class":509,"line":1518},[507,66891,63762],{"class":517},[507,66893,573],{"class":572},[507,66895,58257],{"class":517},[507,66897,63769],{"class":576},[507,66899,1376],{"class":517},[507,66901,66902,66905,66907],{"class":509,"line":1524},[507,66903,66904],{"class":517},"    F2_hi, ",[507,66906,66801],{"class":583},[507,66908,66909],{"class":517},", S_final,\n",[507,66911,66912,66914,66916,66918,66920,66922,66924,66926],{"class":509,"line":1530},[507,66913,63781],{"class":2155},[507,66915,573],{"class":572},[507,66917,625],{"class":583},[507,66919,622],{"class":517},[507,66921,63790],{"class":2155},[507,66923,573],{"class":572},[507,66925,625],{"class":583},[507,66927,1409],{"class":517},[507,66929,66930,66932,66934,66936,66938,66940,66942,66944],{"class":509,"line":1536},[507,66931,63805],{"class":2155},[507,66933,573],{"class":572},[507,66935,601],{"class":583},[507,66937,622],{"class":517},[507,66939,63814],{"class":2155},[507,66941,573],{"class":572},[507,66943,13878],{"class":583},[507,66945,1409],{"class":517},[507,66947,66948,66950,66952,66954],{"class":509,"line":1542},[507,66949,63825],{"class":2155},[507,66951,573],{"class":572},[507,66953,66119],{"class":730},[507,66955,1409],{"class":517},[507,66957,66958,66960,66962,66965,66967,66969],{"class":509,"line":1548},[507,66959,63836],{"class":2155},[507,66961,573],{"class":572},[507,66963,66964],{"class":517},"vmin, ",[507,66966,63845],{"class":2155},[507,66968,573],{"class":572},[507,66970,66971],{"class":517},"vmax\n",[507,66973,66974],{"class":509,"line":1553},[507,66975,587],{"class":517},[507,66977,66978],{"class":509,"line":1559},[507,66979,556],{"emptyLinePlaceholder":133},[507,66981,66982,66984,66986,66988,66991],{"class":509,"line":1565},[507,66983,58314],{"class":517},[507,66985,58317],{"class":576},[507,66987,580],{"class":517},[507,66989,66990],{"class":730},"\"f (GHz)\"",[507,66992,587],{"class":517},[507,66994,66995,66997,66999,67001,67003,67006,67008,67010],{"class":509,"line":1570},[507,66996,58314],{"class":517},[507,66998,58331],{"class":576},[507,67000,580],{"class":517},[507,67002,22278],{"class":513},[507,67004,67005],{"class":730},"\"Fourier freq (",[507,67007,66263],{"class":583},[507,67009,66266],{"class":730},[507,67011,587],{"class":517},[507,67013,67014,67016,67018,67020,67022,67024,67026,67028,67030,67032,67034,67036,67038,67040,67042,67044,67046],{"class":509,"line":1575},[507,67015,58314],{"class":517},[507,67017,63909],{"class":576},[507,67019,580],{"class":517},[507,67021,66152],{"class":730},[507,67023,8313],{"class":572},[507,67025,58644],{"class":517},[507,67027,66159],{"class":730},[507,67029,66162],{"class":513},[507,67031,66165],{"class":583},[507,67033,53],{"class":517},[507,67035,64860],{"class":576},[507,67037,1677],{"class":517},[507,67039,1723],{"class":572},[507,67041,65612],{"class":730},[507,67043,8480],{"class":513},[507,67045,66182],{"class":730},[507,67047,22540],{"class":517},[507,67049,67050,67052,67054,67056,67059],{"class":509,"line":1580},[507,67051,58314],{"class":517},[507,67053,58345],{"class":576},[507,67055,580],{"class":517},[507,67057,67058],{"class":730},"\"Rabi FFT surface (adaptive resolution)\"",[507,67060,587],{"class":517},[507,67062,67063,67065,67067,67069,67071,67073,67075,67077,67079,67081,67083,67085,67087,67089,67091,67093,67095,67098],{"class":509,"line":1597},[507,67064,58367],{"class":517},[507,67066,58286],{"class":576},[507,67068,63952],{"class":517},[507,67070,58292],{"class":2155},[507,67072,573],{"class":572},[507,67074,58297],{"class":517},[507,67076,63961],{"class":2155},[507,67078,573],{"class":572},[507,67080,63966],{"class":583},[507,67082,622],{"class":517},[507,67084,63971],{"class":2155},[507,67086,573],{"class":572},[507,67088,63976],{"class":583},[507,67090,622],{"class":517},[507,67092,21012],{"class":2155},[507,67094,573],{"class":572},[507,67096,67097],{"class":730},"\"Amplitude\"",[507,67099,587],{"class":517},[507,67101,67102],{"class":509,"line":1608},[507,67103,556],{"emptyLinePlaceholder":133},[507,67105,67106],{"class":509,"line":1624},[507,67107,67108],{"class":562},"# A clearer default view\n",[507,67110,67111,67113,67116,67118,67121,67123,67125,67127,67130,67132,67134],{"class":509,"line":1657},[507,67112,58314],{"class":517},[507,67114,67115],{"class":576},"view_init",[507,67117,580],{"class":517},[507,67119,67120],{"class":2155},"elev",[507,67122,573],{"class":572},[507,67124,48588],{"class":583},[507,67126,622],{"class":517},[507,67128,67129],{"class":2155},"azim",[507,67131,14115],{"class":572},[507,67133,58677],{"class":583},[507,67135,587],{"class":517},[507,67137,67138],{"class":509,"line":1663},[507,67139,556],{"emptyLinePlaceholder":133},[507,67141,67142,67144,67146],{"class":509,"line":1691},[507,67143,58367],{"class":517},[507,67145,58370],{"class":576},[507,67147,781],{"class":517},[507,67149,67150,67152,67154],{"class":509,"line":1714},[507,67151,25376],{"class":517},[507,67153,25613],{"class":576},[507,67155,781],{"class":517},[831,67157],{"alt":67158,"src":67159},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[498,67161,67163],{"className":500,"code":67162,"language":502,"meta":104,"style":104},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\nF_STAR_GHZ: float = float(F0_GHZ)   # slice near resonance\nNU_STAR = float(ridge_axis[np.argmin(np.abs(freq_ghz - F_STAR_GHZ))])  # near Ω_R\u002F2π\n\n# Slice at f = F_STAR_GHZ\nj = int(np.argmin(np.abs(freq_ghz - F_STAR_GHZ)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(f_fft_axis, S_fft[:, j], color='blue')\nax.axvline(ridge_axis[j], ls=\"--\", lw=1.0, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at $f={freq_ghz[j]:.6f}$ GHz\")\nax.legend()\nfig.tight_layout(); plt.show()\n\n# Slice at ν = NU_STAR\ni = int(np.argmin(np.abs(f_fft_axis - NU_STAR)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(freq_ghz, S_fft[i, :], color='blue')\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at Fourier freq ≈ {f_fft_axis[i]:.3f} {FREQ_UNITS}\")\nfig.tight_layout(); plt.show()\n",[504,67164,67165,67170,67193,67225,67229,67234,67258,67286,67303,67348,67366,67402,67429,67437,67450,67454,67459,67485,67511,67528,67540,67576,67605],{"__ignoreMap":104},[507,67166,67167],{"class":509,"line":510},[507,67168,67169],{"class":562},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[507,67171,67172,67175,67177,67179,67181,67183,67185,67187,67190],{"class":509,"line":105},[507,67173,67174],{"class":583},"F_STAR_GHZ",[507,67176,1403],{"class":517},[507,67178,1406],{"class":572},[507,67180,1423],{"class":572},[507,67182,59620],{"class":572},[507,67184,580],{"class":517},[507,67186,56527],{"class":583},[507,67188,67189],{"class":517},")   ",[507,67191,67192],{"class":562},"# slice near resonance\n",[507,67194,67195,67198,67200,67202,67205,67207,67209,67211,67214,67216,67219,67222],{"class":509,"line":540},[507,67196,67197],{"class":583},"NU_STAR",[507,67199,1423],{"class":572},[507,67201,59620],{"class":572},[507,67203,67204],{"class":517},"(ridge_axis[np.",[507,67206,64254],{"class":576},[507,67208,59917],{"class":517},[507,67210,64259],{"class":576},[507,67212,67213],{"class":517},"(freq_ghz ",[507,67215,2367],{"class":572},[507,67217,67218],{"class":583}," F_STAR_GHZ",[507,67220,67221],{"class":517},"))])  ",[507,67223,67224],{"class":562},"# near Ω_R\u002F2π\n",[507,67226,67227],{"class":509,"line":553},[507,67228,556],{"emptyLinePlaceholder":133},[507,67230,67231],{"class":509,"line":559},[507,67232,67233],{"class":562},"# Slice at f = F_STAR_GHZ\n",[507,67235,67236,67238,67240,67242,67244,67246,67248,67250,67252,67254,67256],{"class":509,"line":566},[507,67237,64281],{"class":517},[507,67239,573],{"class":572},[507,67241,2095],{"class":572},[507,67243,59917],{"class":517},[507,67245,64254],{"class":576},[507,67247,59917],{"class":517},[507,67249,64259],{"class":576},[507,67251,67213],{"class":517},[507,67253,2367],{"class":572},[507,67255,67218],{"class":583},[507,67257,14066],{"class":517},[507,67259,67260,67262,67264,67266,67268,67270,67272,67274,67276,67279,67281,67284],{"class":509,"line":590},[507,67261,58220],{"class":517},[507,67263,573],{"class":572},[507,67265,58225],{"class":517},[507,67267,58228],{"class":576},[507,67269,580],{"class":517},[507,67271,58233],{"class":2155},[507,67273,573],{"class":572},[507,67275,580],{"class":517},[507,67277,67278],{"class":583},"6.0",[507,67280,622],{"class":517},[507,67282,67283],{"class":583},"3.8",[507,67285,22540],{"class":517},[507,67287,67288,67290,67292,67295,67297,67299,67301],{"class":509,"line":610},[507,67289,58314],{"class":517},[507,67291,25379],{"class":576},[507,67293,67294],{"class":517},"(f_fft_axis, S_fft[:, j], ",[507,67296,64350],{"class":2155},[507,67298,573],{"class":572},[507,67300,64355],{"class":730},[507,67302,587],{"class":517},[507,67304,67305,67307,67310,67313,67316,67318,67320,67322,67324,67326,67328,67330,67332,67334,67336,67338,67340,67342,67344,67346],{"class":509,"line":634},[507,67306,58314],{"class":517},[507,67308,67309],{"class":576},"axvline",[507,67311,67312],{"class":517},"(ridge_axis[j], ",[507,67314,67315],{"class":2155},"ls",[507,67317,573],{"class":572},[507,67319,66200],{"class":730},[507,67321,622],{"class":517},[507,67323,66205],{"class":2155},[507,67325,573],{"class":572},[507,67327,57927],{"class":583},[507,67329,622],{"class":517},[507,67331,21012],{"class":2155},[507,67333,573],{"class":572},[507,67335,2216],{"class":513},[507,67337,61680],{"class":58306},[507,67339,66223],{"class":572},[507,67341,66226],{"class":58306},[507,67343,66229],{"class":572},[507,67345,66232],{"class":58306},[507,67347,587],{"class":517},[507,67349,67350,67352,67354,67356,67358,67360,67362,67364],{"class":509,"line":661},[507,67351,58314],{"class":517},[507,67353,58317],{"class":576},[507,67355,580],{"class":517},[507,67357,22278],{"class":513},[507,67359,66260],{"class":730},[507,67361,66263],{"class":583},[507,67363,66266],{"class":730},[507,67365,587],{"class":517},[507,67367,67368,67370,67372,67374,67376,67378,67380,67382,67384,67386,67388,67390,67392,67394,67396,67398,67400],{"class":509,"line":678},[507,67369,58314],{"class":517},[507,67371,58331],{"class":576},[507,67373,580],{"class":517},[507,67375,66152],{"class":730},[507,67377,8313],{"class":572},[507,67379,58644],{"class":517},[507,67381,66159],{"class":730},[507,67383,66162],{"class":513},[507,67385,66165],{"class":583},[507,67387,53],{"class":517},[507,67389,64860],{"class":576},[507,67391,66172],{"class":517},[507,67393,1723],{"class":572},[507,67395,66177],{"class":730},[507,67397,8480],{"class":513},[507,67399,66182],{"class":730},[507,67401,22540],{"class":517},[507,67403,67404,67406,67408,67410,67413,67416,67418,67420,67422,67424,67427],{"class":509,"line":683},[507,67405,58314],{"class":517},[507,67407,58345],{"class":576},[507,67409,580],{"class":517},[507,67411,67412],{"class":513},"fr",[507,67414,67415],{"class":730},"\"Slice at $f=",[507,67417,2810],{"class":583},[507,67419,64439],{"class":517},[507,67421,62878],{"class":513},[507,67423,2872],{"class":583},[507,67425,67426],{"class":730},"$ GHz\"",[507,67428,587],{"class":517},[507,67430,67431,67433,67435],{"class":509,"line":697},[507,67432,58314],{"class":517},[507,67434,25558],{"class":576},[507,67436,781],{"class":517},[507,67438,67439,67441,67443,67446,67448],{"class":509,"line":710},[507,67440,58367],{"class":517},[507,67442,58370],{"class":576},[507,67444,67445],{"class":517},"(); plt.",[507,67447,25613],{"class":576},[507,67449,781],{"class":517},[507,67451,67452],{"class":509,"line":715},[507,67453,556],{"emptyLinePlaceholder":133},[507,67455,67456],{"class":509,"line":721},[507,67457,67458],{"class":562},"# Slice at ν = NU_STAR\n",[507,67460,67461,67463,67465,67467,67469,67471,67473,67475,67478,67480,67483],{"class":509,"line":736},[507,67462,64495],{"class":517},[507,67464,573],{"class":572},[507,67466,2095],{"class":572},[507,67468,59917],{"class":517},[507,67470,64254],{"class":576},[507,67472,59917],{"class":517},[507,67474,64259],{"class":576},[507,67476,67477],{"class":517},"(f_fft_axis ",[507,67479,2367],{"class":572},[507,67481,67482],{"class":583}," NU_STAR",[507,67484,14066],{"class":517},[507,67486,67487,67489,67491,67493,67495,67497,67499,67501,67503,67505,67507,67509],{"class":509,"line":748},[507,67488,58220],{"class":517},[507,67490,573],{"class":572},[507,67492,58225],{"class":517},[507,67494,58228],{"class":576},[507,67496,580],{"class":517},[507,67498,58233],{"class":2155},[507,67500,573],{"class":572},[507,67502,580],{"class":517},[507,67504,67278],{"class":583},[507,67506,622],{"class":517},[507,67508,67283],{"class":583},[507,67510,22540],{"class":517},[507,67512,67513,67515,67517,67520,67522,67524,67526],{"class":509,"line":761},[507,67514,58314],{"class":517},[507,67516,25379],{"class":576},[507,67518,67519],{"class":517},"(freq_ghz, S_fft[i, :], ",[507,67521,64350],{"class":2155},[507,67523,573],{"class":572},[507,67525,64355],{"class":730},[507,67527,587],{"class":517},[507,67529,67530,67532,67534,67536,67538],{"class":509,"line":775},[507,67531,58314],{"class":517},[507,67533,58317],{"class":576},[507,67535,580],{"class":517},[507,67537,66245],{"class":730},[507,67539,587],{"class":517},[507,67541,67542,67544,67546,67548,67550,67552,67554,67556,67558,67560,67562,67564,67566,67568,67570,67572,67574],{"class":509,"line":784},[507,67543,58314],{"class":517},[507,67545,58331],{"class":576},[507,67547,580],{"class":517},[507,67549,66152],{"class":730},[507,67551,8313],{"class":572},[507,67553,58644],{"class":517},[507,67555,66159],{"class":730},[507,67557,66162],{"class":513},[507,67559,66165],{"class":583},[507,67561,53],{"class":517},[507,67563,64860],{"class":576},[507,67565,66172],{"class":517},[507,67567,1723],{"class":572},[507,67569,66177],{"class":730},[507,67571,8480],{"class":513},[507,67573,66182],{"class":730},[507,67575,22540],{"class":517},[507,67577,67578,67580,67582,67584,67586,67589,67591,67594,67596,67598,67601,67603],{"class":509,"line":796},[507,67579,58314],{"class":517},[507,67581,58345],{"class":576},[507,67583,580],{"class":517},[507,67585,67412],{"class":513},[507,67587,67588],{"class":730},"\"Slice at Fourier freq ≈ ",[507,67590,2810],{"class":583},[507,67592,67593],{"class":517},"f_fft_axis[i]",[507,67595,62953],{"class":513},[507,67597,2872],{"class":583},[507,67599,67600],{"class":583}," {FREQ_UNITS}",[507,67602,22281],{"class":730},[507,67604,587],{"class":517},[507,67606,67607,67609,67611,67613,67615],{"class":509,"line":809},[507,67608,58367],{"class":517},[507,67610,58370],{"class":576},[507,67612,67445],{"class":517},[507,67614,25613],{"class":576},[507,67616,781],{"class":517},[831,67618],{"alt":67619,"src":67620},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[831,67622],{"alt":67623,"src":67624},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[13,67626,940],{"id":939},[18,67628,41598,67629,947,67631,53],{},[49,67630,41670],{"href":41669},[49,67632,67634],{"href":67633},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb","open it in Colab",[953,67636,67637],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: 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.sVyAn{--shiki-default:#E06C75}",{"title":104,"searchDepth":105,"depth":105,"links":67639},[67640,67641,67642,67643,67644,67645,67646,67647,67648,67649,67650,67651,67652,67653,67654],{"id":41846,"depth":105,"text":41847},{"id":43326,"depth":105,"text":43327},{"id":44115,"depth":105,"text":44116},{"id":45464,"depth":105,"text":45465},{"id":46593,"depth":105,"text":46594},{"id":49075,"depth":105,"text":49076},{"id":49591,"depth":105,"text":49592},{"id":50690,"depth":105,"text":50691},{"id":52282,"depth":105,"text":52283},{"id":52655,"depth":105,"text":52656},{"id":53647,"depth":105,"text":53648},{"id":54824,"depth":105,"text":54825},{"id":56004,"depth":105,"text":56005},{"id":56499,"depth":105,"text":56500},{"id":939,"depth":105,"text":940},[112,969,970,67656],"Rabi oscillations",[67658],{"username":467,"name":974,"role":975,"bio":976,"links":67659},[67660,67661],{"label":979,"href":980},{"label":982,"href":41669},{"username":467,"name":974,"role":984},"A primer on the quantum dynamics of a driven two-level system, visualizing excited-state probability under the rotating-wave approximation (RWA).","Deep dive · Hardware",{},"\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[],{"title":41659,"description":67663},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[67672,1019],"hardware","cLyZBkTx0Kr_9zY5k_7c4mZTRRqgWxllLNGfFbxk58w",{"id":67675,"title":67676,"authors":67677,"body":67678,"breadcrumb":72265,"builders":72267,"byline":72272,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":116,"description":72273,"draft":125,"extension":126,"eyebrow":67664,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":988,"lessonCount":116,"meta":72274,"navigation":133,"newsItems":116,"next":116,"ogImage":72275,"order":116,"outcomes":116,"path":72276,"publishDate":992,"readingTime":72277,"related":72278,"relatedProjects":116,"seo":72279,"stem":72280,"tags":72281,"track":116,"trackName":116,"__hash__":72282},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[467],{"type":10,"value":67679,"toc":72261},[67680,67687,72102,72108,72111,72114,72120,72126,72129,72131,72249,72251,72258],[18,67681,67682],{},[1031,67683,1033,67684,41671],{},[49,67685,41670],{"href":67686},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[498,67688,67690],{"className":500,"code":67689,"language":502,"meta":104,"style":104},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n# generates tables and figures, so you can iterate further.\n#\n# Contents created in the current directory:\n#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n#  - summary.csv                                 (summary table)\n#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n#  - fig_required_shots.png                      (required shots for ±ε)\n#  - simulation_config.json                      (run parameters)\n\nimport json\nimport math\nfrom dataclasses import dataclass\nfrom typing import List, Dict, Tuple\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\nimport nbformat as nbf\n\nplt.rcParams['figure.dpi'] = 200\n\n# ---------- Helper math ----------\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n    \"\"\"\n    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n    Returns (low, high). Works for edge cases k=0 or k=n.\n    \"\"\"\n    if n == 0:\n        return (0.0, 1.0)\n    p_hat = k \u002F n\n    denom = 1.0 + (z ** 2) \u002F n\n    center = (p_hat + (z ** 2) \u002F (2 * n)) \u002F denom\n    span = (z * math.sqrt((p_hat * (1 - p_hat) \u002F n) + (z ** 2) \u002F (4 * n ** 2))) \u002F denom\n    return (max(0.0, center - span), min(1.0, center + span))\n\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n    \"\"\"\n    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n    \"\"\"\n    p = min(max(p, 0.0), 1.0)\n    var = p * (1.0 - p)\n    if var == 0.0:\n        return 1\n    n = (z ** 2) * var \u002F (epsilon ** 2)\n    return max(1, int(math.ceil(n)))\n\n\n# ---------- Domain model ----------\n\n@dataclass(frozen=True)\nclass QubitModel:\n    name: str\n    satisfaction_prob: float  # Probability a single shot meets the analytical criterion (bit = 1)\n    description: str\n\n\nDEFAULT_MODELS: List[QubitModel] = [\n    QubitModel(\n        \"Toy working qubit\",\n        0.50,\n        \"Simple demonstrator; about half shots meet the criterion.\"\n    ),\n    QubitModel(\n        \"NISQ qubit\",\n        0.60,\n        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n    ),\n    QubitModel(\n        \"Fault-tolerant (logical)\",\n        0.90,\n        \"Error-corrected logical qubit; high single-shot success probability.\"\n    ),\n    QubitModel(\n        \"Topologically protected\",\n        0.98,\n        \"Intrinsic protection; very high single-shot success probability.\"\n    ),\n    QubitModel(\n        \"Ideal qubit\",\n        1.00,\n        \"Theoretical perfect qubit; always successful in a single shot.\"\n    ),\n]\n\n# ---------- Simulation configuration ----------\n\nCONFIG: Dict = {\n    \"random_seed\": 123456,\n    \"shots_per_model\": 10000,\n    \"epsilons\": [0.02, 0.01, 0.005],  # ±2%, ±1%, ±0.5% targets\n    \"cost_model\": {\n        \"default_cost_per_shot_usd\": 0.00005,\n        \"fixed_overhead_usd\": 0.10\n    },\n    \"timing\": {\n        \"time_per_shot_seconds\": 2e-6\n    }\n}\n\nrng = np.random.default_rng(CONFIG[\"random_seed\"])\n\n\n# ---------- Core simulation ----------\n\ndef simulate_bitstring(p: float, shots: int, rng_: np.random.Generator) -> np.ndarray:\n    \"\"\"\n    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n    \"\"\"\n    return rng_.binomial(1, p, size=shots).astype(int)\n\n\ndef summarize_model(model: QubitModel, bits: np.ndarray, epsilons: List[float]) -> Dict:\n    \"\"\"\n    Calculate summary statistics for a given model and its simulated bitstring.\n    \"\"\"\n    n = int(bits.size)\n    ones = int(bits.sum())\n    zeros = int(n - ones)\n    frac = ones \u002F n if n > 0 else float(\"nan\")\n    lo, hi = wilson_ci(ones, n)\n\n    row = {\n        \"Model\": model.name,\n        \"Description\": model.description,\n        \"Configured p (target)\": model.satisfaction_prob,\n        \"Shots\": n,\n        \"Ones (successes)\": ones,\n        \"Zeros (failures)\": zeros,\n        \"Ones fraction (observed)\": frac,\n        \"Wilson CI 95% low\": lo,\n        \"Wilson CI 95% high\": hi,\n        \"CI width (95%)\": hi - lo,\n    }\n    for eps in epsilons:\n        row[f\"Required shots for ±{int(eps*100)}%\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n    return row\n\n\ndef apply_costs(shots: int, cost_per_shot: float, fixed_overhead: float) -> float:\n    \"\"\"\n    Calculate illustrative cost based on shots and cost model.\n    \"\"\"\n    return fixed_overhead + shots * cost_per_shot\n\n\n# ---------- Run the simulation ----------\n\nall_bitstrings: Dict[str, np.ndarray] = {}\nsummary_rows: List[Dict] = []\n\nprint(\"Running simulation...\")\nfor m in DEFAULT_MODELS:\n    bits = simulate_bitstring(m.satisfaction_prob, CONFIG[\"shots_per_model\"], rng)\n    all_bitstrings[m.name] = bits\n    summary_rows.append(summarize_model(m, bits, CONFIG[\"epsilons\"]))\n\nsummary_df = pd.DataFrame(summary_rows)\n\n# Add cost estimates\ncps = CONFIG[\"cost_model\"][\"default_cost_per_shot_usd\"]\nover = CONFIG[\"cost_model\"][\"fixed_overhead_usd\"]\nif cps > 0 or over > 0:\n    summary_df[\"Illustrative cost (USD)\"] = [\n        apply_costs(int(row[\"Shots\"]), cps, over)\n        for _, row in summary_df.iterrows()\n    ]\n\n# ---------- Save artifacts ----------\n\n# Save raw bitstrings and summary CSV\nfor name, arr in all_bitstrings.items():\n    out_path = f\"bitstrings_{name.replace(' ', '_')}.csv\"\n    pd.DataFrame({\"bit\": arr}).to_csv(out_path, index=False)\n\nsummary_csv_path = \"summary.csv\"\nsummary_df.to_csv(summary_csv_path, index=False)\n\n# Save config JSON\nwith open(\"simulation_config.json\", \"w\") as f:\n    json.dump(CONFIG, f, indent=2)\n\nprint(\"Saved data to CSV and JSON files.\")\n\n# ---------- Visualizations ----------\n\n# Figure 1: Observed ones fraction with Wilson CI\nplt.style.use('seaborn-v0_8-whitegrid')\nplt.figure(figsize=(10, 6))\nx = np.arange(len(summary_df))\ny = summary_df[\"Ones fraction (observed)\"].to_numpy()\n\n# --- FIX APPLIED HERE ---\n# Take the absolute value to prevent tiny negative numbers from floating-point\n# inaccuracies from causing a ValueError in the errorbar plot.\nlower_error = y - summary_df[\"Wilson CI 95% low\"].to_numpy()\nupper_error = summary_df[\"Wilson CI 95% high\"].to_numpy() - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n# --- END OF FIX ---\n\nplt.bar(x, y, color='skyblue', edgecolor='black')\nplt.errorbar(x, y, yerr=yerr, fmt=\"none\", capsize=5, color=\"black\", elinewidth=1.5)\nplt.xticks(x, summary_df[\"Model\"].tolist(), rotation=25, ha=\"right\")\nplt.ylabel(\"Observed Ones Fraction\")\nplt.title(\"Observed Single-Shot Success Rates with 95% Wilson CI\", fontsize=14)\nplt.tight_layout()\nfig1_path = \"fig_ones_fraction.png\"\nplt.savefig(fig1_path, dpi=200, bbox_inches='tight')\nplt.show()\n\n# Figure 2: Required shots at several ±epsilon targets\nplt.figure(figsize=(10, 6))\nwidth = 0.2\nx = np.arange(len(DEFAULT_MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\n\nfor i, eps in enumerate(CONFIG[\"epsilons\"]):\n    req = [shots_for_epsilon(m.satisfaction_prob, eps) for m in DEFAULT_MODELS]\n    plt.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nplt.xticks(x + width, [m.name for m in DEFAULT_MODELS], rotation=25, ha=\"right\")\nplt.ylabel(\"Required Shots (Log Scale)\")\nplt.title(\"Approximate Shots to Estimate Success Rate within ±ε\", fontsize=14)\nplt.yscale('log')\nplt.legend(title=\"Error Tolerance\")\nplt.tight_layout()\nfig2_path = \"fig_required_shots.png\"\nplt.savefig(fig2_path, dpi=200, bbox_inches='tight')\nplt.show()\n\nprint(\"Generated and saved plots.\")\n\n# ---------- Display interactive table ----------\n\ndisplay(Markdown(\"##  Qubit Sampling Budget - Summary\"))\ndisplay(summary_df.style.format({\n    \"Configured p (target)\": \"{:.2f}\",\n    \"Ones fraction (observed)\": \"{:.4f}\",\n    \"Wilson CI 95% low\": \"{:.4f}\",\n    \"Wilson CI 95% high\": \"{:.4f}\",\n    \"CI width (95%)\": \"{:.4f}\",\n    \"Illustrative cost (USD)\": \"${:.2f}\"\n}))\n\n\n# ---------- Build a ready-to-run Colab notebook ----------\n\nmd_intro = r\"\"\"\n# Sampling Budget & Bit-String Analogy for Qubit Classes\n\n**High school level (concise):** We treat each measurement (“single shot”) as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n\n**Graduate level (concise):** We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ (noisy intermediate-scale quantum), fault-tolerant (logical), topologically protected, and ideal. We report Wilson score intervals (95%) for observed proportions and estimate the required shot budget $n \\approx z^2\\,p(1-p)\u002F\\epsilon^2$ to bound absolute error by ±$\\epsilon$ at ~95% confidence.\n\"\"\"\n\ncode_config = r\"\"\"\n# @title Configuration\nfrom dataclasses import dataclass\nfrom typing import List, Dict\nimport numpy as np\n\nRANDOM_SEED = 579345  # @param {type:\"number\"}\nSHOTS_PER_MODEL = 10000  # @param {type:\"number\"}\nEPSILONS = [0.02, 0.01, 0.005]  # @param\nCOST_PER_SHOT_USD = 0.00005  # @param {type:\"number\"}\nFIXED_OVERHEAD_USD = 0.10    # @param {type:\"number\"}\n\n@dataclass(frozen=True)\nclass QubitModel:\n    name: str\n    satisfaction_prob: float\n    description: str\n\nMODELS: List[QubitModel] = [\n    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n]\n\nrng = np.random.default_rng(RANDOM_SEED)\n\"\"\"\n\ncode_lib_and_run = r\"\"\"\n# @title Run Simulation & Display Results\nimport math\nfrom typing import Tuple, Dict\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n    if n == 0: return (0.0, 1.0)\n    p_hat, z2_n = k \u002F n, z**2 \u002F n\n    denom = 1.0 + z2_n\n    center = (p_hat + z2_n \u002F 2) \u002F denom\n    span = (z * math.sqrt(p_hat * (1 - p_hat) \u002F n + z**2 \u002F (4 * n**2))) \u002F denom\n    return (max(0.0, center - span), min(1.0, center + span))\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n    if p * (1-p) == 0: return 1\n    return max(1, int(math.ceil((z**2 * p * (1-p)) \u002F (epsilon**2))))\n\ndef summarize_model(model, bits: np.ndarray, epsilons) -> Dict:\n    n = len(bits)\n    ones = sum(bits)\n    lo, hi = wilson_ci(ones, n)\n    row = {\n        \"Model\": model.name, \"p (target)\": model.satisfaction_prob, \"Shots\": n,\n        \"Ones\": ones, \"p (observed)\": ones \u002F n if n > 0 else 0,\n        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n    }\n    for eps in epsilons:\n        row[f\"Shots for ±{eps:.1%}\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n    return row\n\n# --- Run Simulation ---\nsummary_rows = [summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\nsummary_df = pd.DataFrame(summary_rows)\nif COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n    summary_df[\"Cost (USD)\"] = FIXED_OVERHEAD_USD + summary_df[\"Shots\"] * COST_PER_SHOT_USD\n\ndisplay(Markdown(\"### Simulation Summary\"))\ndisplay(summary_df.style.format(precision=4).background_gradient(cmap='viridis', subset=['p (observed)', 'CI width']))\n\n# --- Plotting ---\nplt.style.use('seaborn-v0_8-whitegrid')\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))\n\n# Plot 1\ny = summary_df[\"p (observed)\"].to_numpy()\nlower_error = y - summary_df[\"CI 95% low\"]\nupper_error = summary_df[\"CI 95% high\"] - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n\nax1.bar(summary_df[\"Model\"], y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black')\nax1.set_title(\"Observed Success Rate (95% Wilson CI)\", fontsize=14)\nax1.tick_params(axis='x', rotation=30, labelsize=10)\n\n# Plot 2\nwidth = 0.2\nx = np.arange(len(MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\nfor i, eps in enumerate(EPSILONS):\n    req = summary_df[f\"Shots for ±{eps:.1%}\"]\n    ax2.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nax2.set_xticks(x + width, summary_df[\"Model\"])\nax2.set_ylabel(\"Required Shots (Log Scale)\")\nax2.set_yscale('log')\nax2.set_title(\"Shots Needed for ±ε Tolerance\", fontsize=14)\nax2.tick_params(axis='x', rotation=30, labelsize=10)\nax2.legend(title=\"Tolerance\")\n\nplt.tight_layout()\nplt.show()\n\"\"\"\n\nnb = nbf.v4.new_notebook()\nnb[\"cells\"] = [\n    nbf.v4.new_markdown_cell(md_intro),\n    nbf.v4.new_code_cell(code_config),\n    nbf.v4.new_code_cell(code_lib_and_run),\n]\n\nnb_path = \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\nwith open(nb_path, \"w\", encoding=\"utf-8\") as f:\n    nbf.write(nb, f)\n\n# ---------- Final confirmation ----------\nprint(\"\\n\" + \"=\"*50)\nprint(\"SCRIPT COMPLETE\")\nprint(\"=\"*50)\nprint(f\"Google Colab notebook saved to: {nb_path}\")\nprint(f\"Summary data saved to: {summary_csv_path}\")\nprint(f\"Plots saved to: {fig1_path}, {fig2_path}\")\n",[504,67691,67692,67697,67702,67707,67712,67717,67722,67727,67732,67737,67742,67746,67753,67759,67771,67782,67786,67796,67806,67816,67828,67840,67844,67857,67861,67866,67870,67917,67921,67926,67931,67935,67948,67962,67976,68001,68037,68103,68136,68140,68144,68191,68195,68200,68205,68209,68233,68254,68267,68273,68303,68325,68329,68333,68338,68342,68358,68369,68377,68387,68394,68398,68402,68414,68421,68428,68435,68440,68444,68450,68457,68464,68469,68473,68479,68486,68493,68498,68502,68508,68515,68522,68527,68531,68537,68544,68551,68556,68560,68564,68568,68573,68577,68589,68601,68613,68639,68646,68658,68668,68672,68679,68689,68693,68697,68701,68726,68730,68734,68739,68743,68775,68779,68784,68788,68821,68825,68829,68860,68864,68869,68873,68884,68901,68918,68951,68963,68967,68976,68984,68992,69000,69008,69016,69024,69032,69046,69059,69072,69076,69088,69125,69132,69136,69140,69179,69183,69188,69192,69209,69213,69217,69222,69226,69240,69249,69253,69264,69278,69300,69310,69335,69339,69353,69357,69362,69384,69404,69426,69440,69458,69475,69479,69483,69488,69492,69497,69513,69550,69581,69585,69595,69613,69617,69622,69647,69671,69675,69686,69690,69695,69699,69704,69719,69741,69760,69780,69784,69789,69794,69799,69826,69852,69876,69881,69885,69914,69968,70005,70018,70041,70049,70059,70087,70095,70099,70104,70126,70136,70156,70180,70184,70206,70231,70288,70292,70331,70344,70365,70379,70396,70404,70414,70439,70447,70451,70462,70466,70471,70475,70492,70505,70525,70545,70568,70590,70609,70626,70631,70635,70639,70644,70648,70661,70666,70670,70701,70705,70786,70790,70794,70805,70810,70815,70820,70825,70829,70837,70844,70855,70862,70869,70873,70884,70889,70894,70899,70904,70908,70922,70927,70932,70937,70942,70947,70951,70955,70967,70971,70975,70986,70991,70996,71001,71006,71010,71015,71020,71024,71044,71056,71066,71076,71096,71156,71187,71191,71206,71222,71273,71277,71292,71303,71314,71326,71331,71346,71361,71366,71370,71375,71393,71398,71402,71407,71415,71427,71432,71456,71460,71475,71502,71506,71511,71522,71549,71553,71558,71571,71579,71590,71598,71602,71620,71642,71654,71658,71663,71668,71684,71692,71706,71714,71740,71744,71761,71782,71793,71805,71816,71828,71832,71839,71846,71850,71854,71869,71883,71894,71904,71913,71917,71921,71931,71958,71969,71973,71978,72001,72012,72027,72049,72071],{"__ignoreMap":104},[507,67693,67694],{"class":509,"line":510},[507,67695,67696],{"class":562},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[507,67698,67699],{"class":509,"line":105},[507,67700,67701],{"class":562},"# generates tables and figures, so you can iterate further.\n",[507,67703,67704],{"class":509,"line":540},[507,67705,67706],{"class":562},"#\n",[507,67708,67709],{"class":509,"line":553},[507,67710,67711],{"class":562},"# Contents created in the current directory:\n",[507,67713,67714],{"class":509,"line":559},[507,67715,67716],{"class":562},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[507,67718,67719],{"class":509,"line":566},[507,67720,67721],{"class":562},"#  - summary.csv                                 (summary table)\n",[507,67723,67724],{"class":509,"line":590},[507,67725,67726],{"class":562},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[507,67728,67729],{"class":509,"line":610},[507,67730,67731],{"class":562},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[507,67733,67734],{"class":509,"line":634},[507,67735,67736],{"class":562},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[507,67738,67739],{"class":509,"line":661},[507,67740,67741],{"class":562},"#  - simulation_config.json                      (run parameters)\n",[507,67743,67744],{"class":509,"line":678},[507,67745,556],{"emptyLinePlaceholder":133},[507,67747,67748,67750],{"class":509,"line":683},[507,67749,514],{"class":513},[507,67751,67752],{"class":517}," json\n",[507,67754,67755,67757],{"class":509,"line":697},[507,67756,514],{"class":513},[507,67758,57135],{"class":517},[507,67760,67761,67763,67766,67768],{"class":509,"line":710},[507,67762,529],{"class":513},[507,67764,67765],{"class":517}," dataclasses ",[507,67767,514],{"class":513},[507,67769,67770],{"class":517}," dataclass\n",[507,67772,67773,67775,67777,67779],{"class":509,"line":715},[507,67774,529],{"class":513},[507,67776,1166],{"class":517},[507,67778,514],{"class":513},[507,67780,67781],{"class":517}," List, Dict, Tuple\n",[507,67783,67784],{"class":509,"line":721},[507,67785,556],{"emptyLinePlaceholder":133},[507,67787,67788,67790,67792,67794],{"class":509,"line":736},[507,67789,514],{"class":513},[507,67791,518],{"class":517},[507,67793,521],{"class":513},[507,67795,524],{"class":517},[507,67797,67798,67800,67802,67804],{"class":509,"line":748},[507,67799,514],{"class":513},[507,67801,57174],{"class":517},[507,67803,521],{"class":513},[507,67805,57179],{"class":517},[507,67807,67808,67810,67812,67814],{"class":509,"line":761},[507,67809,514],{"class":513},[507,67811,57163],{"class":517},[507,67813,521],{"class":513},[507,67815,1159],{"class":517},[507,67817,67818,67820,67823,67825],{"class":509,"line":775},[507,67819,529],{"class":513},[507,67821,67822],{"class":517}," IPython.display ",[507,67824,514],{"class":513},[507,67826,67827],{"class":517}," display, Markdown\n",[507,67829,67830,67832,67835,67837],{"class":509,"line":784},[507,67831,514],{"class":513},[507,67833,67834],{"class":517}," nbformat ",[507,67836,521],{"class":513},[507,67838,67839],{"class":517}," nbf\n",[507,67841,67842],{"class":509,"line":796},[507,67843,556],{"emptyLinePlaceholder":133},[507,67845,67846,67848,67850,67852,67854],{"class":509,"line":809},[507,67847,57242],{"class":517},[507,67849,57245],{"class":730},[507,67851,8206],{"class":517},[507,67853,573],{"class":572},[507,67855,67856],{"class":583}," 200\n",[507,67858,67859],{"class":509,"line":1352},[507,67860,556],{"emptyLinePlaceholder":133},[507,67862,67863],{"class":509,"line":1357},[507,67864,67865],{"class":562},"# ---------- Helper math ----------\n",[507,67867,67868],{"class":509,"line":1362},[507,67869,556],{"emptyLinePlaceholder":133},[507,67871,67872,67874,67877,67879,67881,67883,67885,67887,67889,67891,67893,67895,67897,67899,67901,67903,67906,67909,67911,67913,67915],{"class":509,"line":1367},[507,67873,1370],{"class":513},[507,67875,67876],{"class":576}," wilson_ci",[507,67878,580],{"class":517},[507,67880,3626],{"class":1382},[507,67882,1403],{"class":517},[507,67884,1420],{"class":572},[507,67886,622],{"class":517},[507,67888,4420],{"class":1382},[507,67890,1403],{"class":517},[507,67892,1420],{"class":572},[507,67894,622],{"class":517},[507,67896,666],{"class":1382},[507,67898,1403],{"class":517},[507,67900,1406],{"class":572},[507,67902,1423],{"class":572},[507,67904,67905],{"class":583}," 1.96",[507,67907,67908],{"class":517},") -> Tuple[",[507,67910,1406],{"class":572},[507,67912,622],{"class":517},[507,67914,1406],{"class":572},[507,67916,59623],{"class":517},[507,67918,67919],{"class":509,"line":1379},[507,67920,1468],{"class":730},[507,67922,67923],{"class":509,"line":1389},[507,67924,67925],{"class":730},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[507,67927,67928],{"class":509,"line":1397},[507,67929,67930],{"class":730},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[507,67932,67933],{"class":509,"line":1412},[507,67934,1468],{"class":730},[507,67936,67937,67939,67942,67944,67946],{"class":509,"line":1431},[507,67938,1717],{"class":513},[507,67940,67941],{"class":517}," n ",[507,67943,1723],{"class":572},[507,67945,21258],{"class":583},[507,67947,1728],{"class":517},[507,67949,67950,67952,67954,67956,67958,67960],{"class":509,"line":1449},[507,67951,64873],{"class":513},[507,67953,58644],{"class":517},[507,67955,56714],{"class":583},[507,67957,622],{"class":517},[507,67959,57927],{"class":583},[507,67961,587],{"class":517},[507,67963,67964,67967,67969,67971,67973],{"class":509,"line":1465},[507,67965,67966],{"class":517},"    p_hat ",[507,67968,573],{"class":572},[507,67970,1720],{"class":517},[507,67972,645],{"class":572},[507,67974,67975],{"class":517}," n\n",[507,67977,67978,67981,67983,67986,67988,67991,67993,67995,67997,67999],{"class":509,"line":1471},[507,67979,67980],{"class":517},"    denom ",[507,67982,573],{"class":572},[507,67984,67985],{"class":583}," 1.0",[507,67987,8313],{"class":572},[507,67989,67990],{"class":517}," (z ",[507,67992,2377],{"class":572},[507,67994,2316],{"class":583},[507,67996,655],{"class":517},[507,67998,645],{"class":572},[507,68000,67975],{"class":517},[507,68002,68003,68006,68008,68011,68013,68015,68017,68019,68021,68023,68025,68027,68029,68032,68034],{"class":509,"line":1477},[507,68004,68005],{"class":517},"    center ",[507,68007,573],{"class":572},[507,68009,68010],{"class":517}," (p_hat ",[507,68012,2107],{"class":572},[507,68014,67990],{"class":517},[507,68016,2377],{"class":572},[507,68018,2316],{"class":583},[507,68020,655],{"class":517},[507,68022,645],{"class":572},[507,68024,58644],{"class":517},[507,68026,584],{"class":583},[507,68028,8229],{"class":572},[507,68030,68031],{"class":517}," n)) ",[507,68033,645],{"class":572},[507,68035,68036],{"class":517}," denom\n",[507,68038,68039,68042,68044,68046,68048,68051,68053,68056,68058,68060,68062,68064,68067,68069,68072,68074,68076,68078,68080,68082,68084,68086,68088,68090,68092,68094,68096,68099,68101],{"class":509,"line":1482},[507,68040,68041],{"class":517},"    span ",[507,68043,573],{"class":572},[507,68045,67990],{"class":517},[507,68047,2391],{"class":572},[507,68049,68050],{"class":517}," math.",[507,68052,9819],{"class":576},[507,68054,68055],{"class":517},"((p_hat ",[507,68057,2391],{"class":572},[507,68059,58644],{"class":517},[507,68061,625],{"class":583},[507,68063,65914],{"class":572},[507,68065,68066],{"class":517}," p_hat) ",[507,68068,645],{"class":572},[507,68070,68071],{"class":517}," n) ",[507,68073,2107],{"class":572},[507,68075,67990],{"class":517},[507,68077,2377],{"class":572},[507,68079,2316],{"class":583},[507,68081,655],{"class":517},[507,68083,645],{"class":572},[507,68085,58644],{"class":517},[507,68087,12152],{"class":583},[507,68089,8229],{"class":572},[507,68091,67941],{"class":517},[507,68093,2377],{"class":572},[507,68095,2316],{"class":583},[507,68097,68098],{"class":517},"))) ",[507,68100,645],{"class":572},[507,68102,68036],{"class":517},[507,68104,68105,68107,68109,68111,68113,68115,68118,68120,68123,68125,68127,68129,68131,68133],{"class":509,"line":1488},[507,68106,2504],{"class":513},[507,68108,58644],{"class":517},[507,68110,36712],{"class":572},[507,68112,580],{"class":517},[507,68114,56714],{"class":583},[507,68116,68117],{"class":517},", center ",[507,68119,2367],{"class":572},[507,68121,68122],{"class":517}," span), ",[507,68124,25230],{"class":572},[507,68126,580],{"class":517},[507,68128,57927],{"class":583},[507,68130,68117],{"class":517},[507,68132,2107],{"class":572},[507,68134,68135],{"class":517}," span))\n",[507,68137,68138],{"class":509,"line":1494},[507,68139,556],{"emptyLinePlaceholder":133},[507,68141,68142],{"class":509,"line":1500},[507,68143,556],{"emptyLinePlaceholder":133},[507,68145,68146,68148,68151,68153,68155,68157,68159,68161,68164,68166,68168,68170,68173,68175,68177,68179,68181,68183,68185,68187,68189],{"class":509,"line":1506},[507,68147,1370],{"class":513},[507,68149,68150],{"class":576}," shots_for_epsilon",[507,68152,580],{"class":517},[507,68154,18],{"class":1382},[507,68156,1403],{"class":517},[507,68158,1406],{"class":572},[507,68160,622],{"class":517},[507,68162,68163],{"class":1382},"epsilon",[507,68165,1403],{"class":517},[507,68167,1406],{"class":572},[507,68169,1423],{"class":572},[507,68171,68172],{"class":583}," 0.01",[507,68174,622],{"class":517},[507,68176,666],{"class":1382},[507,68178,1403],{"class":517},[507,68180,1406],{"class":572},[507,68182,1423],{"class":572},[507,68184,67905],{"class":583},[507,68186,58900],{"class":517},[507,68188,1420],{"class":572},[507,68190,1728],{"class":517},[507,68192,68193],{"class":509,"line":1512},[507,68194,1468],{"class":730},[507,68196,68197],{"class":509,"line":1518},[507,68198,68199],{"class":730},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[507,68201,68202],{"class":509,"line":1524},[507,68203,68204],{"class":730},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[507,68206,68207],{"class":509,"line":1530},[507,68208,1468],{"class":730},[507,68210,68211,68214,68216,68218,68220,68222,68225,68227,68229,68231],{"class":509,"line":1536},[507,68212,68213],{"class":517},"    p ",[507,68215,573],{"class":572},[507,68217,2476],{"class":572},[507,68219,580],{"class":517},[507,68221,36712],{"class":572},[507,68223,68224],{"class":517},"(p, ",[507,68226,56714],{"class":583},[507,68228,2213],{"class":517},[507,68230,57927],{"class":583},[507,68232,587],{"class":517},[507,68234,68235,68238,68240,68243,68245,68247,68249,68251],{"class":509,"line":1542},[507,68236,68237],{"class":517},"    var ",[507,68239,573],{"class":572},[507,68241,68242],{"class":517}," p ",[507,68244,2391],{"class":572},[507,68246,58644],{"class":517},[507,68248,57927],{"class":583},[507,68250,65914],{"class":572},[507,68252,68253],{"class":517}," p)\n",[507,68255,68256,68258,68261,68263,68265],{"class":509,"line":1548},[507,68257,1717],{"class":513},[507,68259,68260],{"class":517}," var ",[507,68262,1723],{"class":572},[507,68264,57367],{"class":583},[507,68266,1728],{"class":517},[507,68268,68269,68271],{"class":509,"line":1553},[507,68270,64873],{"class":513},[507,68272,2084],{"class":583},[507,68274,68275,68278,68280,68282,68284,68286,68288,68290,68292,68294,68297,68299,68301],{"class":509,"line":1559},[507,68276,68277],{"class":517},"    n ",[507,68279,573],{"class":572},[507,68281,67990],{"class":517},[507,68283,2377],{"class":572},[507,68285,2316],{"class":583},[507,68287,655],{"class":517},[507,68289,2391],{"class":572},[507,68291,68260],{"class":517},[507,68293,645],{"class":572},[507,68295,68296],{"class":517}," (epsilon ",[507,68298,2377],{"class":572},[507,68300,2316],{"class":583},[507,68302,587],{"class":517},[507,68304,68305,68307,68309,68311,68313,68315,68317,68320,68322],{"class":509,"line":1565},[507,68306,2504],{"class":513},[507,68308,65241],{"class":572},[507,68310,580],{"class":517},[507,68312,625],{"class":583},[507,68314,622],{"class":517},[507,68316,1420],{"class":572},[507,68318,68319],{"class":517},"(math.",[507,68321,65228],{"class":576},[507,68323,68324],{"class":517},"(n)))\n",[507,68326,68327],{"class":509,"line":1570},[507,68328,556],{"emptyLinePlaceholder":133},[507,68330,68331],{"class":509,"line":1575},[507,68332,556],{"emptyLinePlaceholder":133},[507,68334,68335],{"class":509,"line":1580},[507,68336,68337],{"class":562},"# ---------- Domain model ----------\n",[507,68339,68340],{"class":509,"line":1597},[507,68341,556],{"emptyLinePlaceholder":133},[507,68343,68344,68347,68349,68352,68354,68356],{"class":509,"line":1608},[507,68345,68346],{"class":576},"@dataclass",[507,68348,580],{"class":517},[507,68350,68351],{"class":2155},"frozen",[507,68353,573],{"class":572},[507,68355,13878],{"class":583},[507,68357,587],{"class":517},[507,68359,68360,68363,68367],{"class":509,"line":1624},[507,68361,68362],{"class":513},"class",[507,68364,68366],{"class":68365},"sU0A5"," QubitModel",[507,68368,1728],{"class":517},[507,68370,68371,68374],{"class":509,"line":1657},[507,68372,68373],{"class":517},"    name: ",[507,68375,68376],{"class":572},"str\n",[507,68378,68379,68382,68384],{"class":509,"line":1663},[507,68380,68381],{"class":517},"    satisfaction_prob: ",[507,68383,1406],{"class":572},[507,68385,68386],{"class":562},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[507,68388,68389,68392],{"class":509,"line":1691},[507,68390,68391],{"class":517},"    description: ",[507,68393,68376],{"class":572},[507,68395,68396],{"class":509,"line":1714},[507,68397,556],{"emptyLinePlaceholder":133},[507,68399,68400],{"class":509,"line":1731},[507,68401,556],{"emptyLinePlaceholder":133},[507,68403,68404,68407,68410,68412],{"class":509,"line":1740},[507,68405,68406],{"class":583},"DEFAULT_MODELS",[507,68408,68409],{"class":517},": List[QubitModel] ",[507,68411,573],{"class":572},[507,68413,2177],{"class":517},[507,68415,68416,68419],{"class":509,"line":1769},[507,68417,68418],{"class":576},"    QubitModel",[507,68420,1376],{"class":517},[507,68422,68423,68426],{"class":509,"line":1777},[507,68424,68425],{"class":730},"        \"Toy working qubit\"",[507,68427,1409],{"class":517},[507,68429,68430,68433],{"class":509,"line":1797},[507,68431,68432],{"class":583},"        0.50",[507,68434,1409],{"class":517},[507,68436,68437],{"class":509,"line":1805},[507,68438,68439],{"class":730},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[507,68441,68442],{"class":509,"line":1812},[507,68443,22632],{"class":517},[507,68445,68446,68448],{"class":509,"line":1832},[507,68447,68418],{"class":576},[507,68449,1376],{"class":517},[507,68451,68452,68455],{"class":509,"line":1839},[507,68453,68454],{"class":730},"        \"NISQ qubit\"",[507,68456,1409],{"class":517},[507,68458,68459,68462],{"class":509,"line":1855},[507,68460,68461],{"class":583},"        0.60",[507,68463,1409],{"class":517},[507,68465,68466],{"class":509,"line":1860},[507,68467,68468],{"class":730},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[507,68470,68471],{"class":509,"line":1865},[507,68472,22632],{"class":517},[507,68474,68475,68477],{"class":509,"line":1886},[507,68476,68418],{"class":576},[507,68478,1376],{"class":517},[507,68480,68481,68484],{"class":509,"line":1891},[507,68482,68483],{"class":730},"        \"Fault-tolerant (logical)\"",[507,68485,1409],{"class":517},[507,68487,68488,68491],{"class":509,"line":1897},[507,68489,68490],{"class":583},"        0.90",[507,68492,1409],{"class":517},[507,68494,68495],{"class":509,"line":1902},[507,68496,68497],{"class":730},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[507,68499,68500],{"class":509,"line":1913},[507,68501,22632],{"class":517},[507,68503,68504,68506],{"class":509,"line":1933},[507,68505,68418],{"class":576},[507,68507,1376],{"class":517},[507,68509,68510,68513],{"class":509,"line":1973},[507,68511,68512],{"class":730},"        \"Topologically protected\"",[507,68514,1409],{"class":517},[507,68516,68517,68520],{"class":509,"line":1978},[507,68518,68519],{"class":583},"        0.98",[507,68521,1409],{"class":517},[507,68523,68524],{"class":509,"line":1988},[507,68525,68526],{"class":730},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[507,68528,68529],{"class":509,"line":2003},[507,68530,22632],{"class":517},[507,68532,68533,68535],{"class":509,"line":2036},[507,68534,68418],{"class":576},[507,68536,1376],{"class":517},[507,68538,68539,68542],{"class":509,"line":2041},[507,68540,68541],{"class":730},"        \"Ideal qubit\"",[507,68543,1409],{"class":517},[507,68545,68546,68549],{"class":509,"line":2052},[507,68547,68548],{"class":583},"        1.00",[507,68550,1409],{"class":517},[507,68552,68553],{"class":509,"line":2057},[507,68554,68555],{"class":730},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[507,68557,68558],{"class":509,"line":2071},[507,68559,22632],{"class":517},[507,68561,68562],{"class":509,"line":2076},[507,68563,1794],{"class":517},[507,68565,68566],{"class":509,"line":2087},[507,68567,556],{"emptyLinePlaceholder":133},[507,68569,68570],{"class":509,"line":2115},[507,68571,68572],{"class":562},"# ---------- Simulation configuration ----------\n",[507,68574,68575],{"class":509,"line":2134},[507,68576,556],{"emptyLinePlaceholder":133},[507,68578,68579,68582,68585,68587],{"class":509,"line":2139},[507,68580,68581],{"class":583},"CONFIG",[507,68583,68584],{"class":517},": Dict ",[507,68586,573],{"class":572},[507,68588,23723],{"class":517},[507,68590,68591,68594,68596,68599],{"class":509,"line":2164},[507,68592,68593],{"class":730},"    \"random_seed\"",[507,68595,1403],{"class":517},[507,68597,68598],{"class":583},"123456",[507,68600,1409],{"class":517},[507,68602,68603,68606,68608,68611],{"class":509,"line":2169},[507,68604,68605],{"class":730},"    \"shots_per_model\"",[507,68607,1403],{"class":517},[507,68609,68610],{"class":583},"10000",[507,68612,1409],{"class":517},[507,68614,68615,68618,68620,68623,68625,68628,68630,68633,68636],{"class":509,"line":2180},[507,68616,68617],{"class":730},"    \"epsilons\"",[507,68619,23857],{"class":517},[507,68621,68622],{"class":583},"0.02",[507,68624,622],{"class":517},[507,68626,68627],{"class":583},"0.01",[507,68629,622],{"class":517},[507,68631,68632],{"class":583},"0.005",[507,68634,68635],{"class":517},"],  ",[507,68637,68638],{"class":562},"# ±2%, ±1%, ±0.5% targets\n",[507,68640,68641,68644],{"class":509,"line":2224},[507,68642,68643],{"class":730},"    \"cost_model\"",[507,68645,23742],{"class":517},[507,68647,68648,68651,68653,68656],{"class":509,"line":2233},[507,68649,68650],{"class":730},"        \"default_cost_per_shot_usd\"",[507,68652,1403],{"class":517},[507,68654,68655],{"class":583},"0.00005",[507,68657,1409],{"class":517},[507,68659,68660,68663,68665],{"class":509,"line":2238},[507,68661,68662],{"class":730},"        \"fixed_overhead_usd\"",[507,68664,1403],{"class":517},[507,68666,68667],{"class":583},"0.10\n",[507,68669,68670],{"class":509,"line":2249},[507,68671,23763],{"class":517},[507,68673,68674,68677],{"class":509,"line":2264},[507,68675,68676],{"class":730},"    \"timing\"",[507,68678,23742],{"class":517},[507,68680,68681,68684,68686],{"class":509,"line":2279},[507,68682,68683],{"class":730},"        \"time_per_shot_seconds\"",[507,68685,1403],{"class":517},[507,68687,68688],{"class":583},"2e-6\n",[507,68690,68691],{"class":509,"line":2290},[507,68692,59483],{"class":517},[507,68694,68695],{"class":509,"line":2295},[507,68696,23875],{"class":517},[507,68698,68699],{"class":509,"line":2321},[507,68700,556],{"emptyLinePlaceholder":133},[507,68702,68703,68706,68708,68711,68714,68716,68718,68720,68723],{"class":509,"line":2337},[507,68704,68705],{"class":517},"rng ",[507,68707,573],{"class":572},[507,68709,68710],{"class":517}," np.random.",[507,68712,68713],{"class":576},"default_rng",[507,68715,580],{"class":517},[507,68717,68581],{"class":583},[507,68719,12248],{"class":517},[507,68721,68722],{"class":730},"\"random_seed\"",[507,68724,68725],{"class":517},"])\n",[507,68727,68728],{"class":509,"line":2356},[507,68729,556],{"emptyLinePlaceholder":133},[507,68731,68732],{"class":509,"line":2396},[507,68733,556],{"emptyLinePlaceholder":133},[507,68735,68736],{"class":509,"line":2419},[507,68737,68738],{"class":562},"# ---------- Core simulation ----------\n",[507,68740,68741],{"class":509,"line":2425},[507,68742,556],{"emptyLinePlaceholder":133},[507,68744,68745,68747,68750,68752,68754,68756,68758,68760,68763,68765,68767,68769,68772],{"class":509,"line":2433},[507,68746,1370],{"class":513},[507,68748,68749],{"class":576}," simulate_bitstring",[507,68751,580],{"class":517},[507,68753,18],{"class":1382},[507,68755,1403],{"class":517},[507,68757,1406],{"class":572},[507,68759,622],{"class":517},[507,68761,68762],{"class":1382},"shots",[507,68764,1403],{"class":517},[507,68766,1420],{"class":572},[507,68768,622],{"class":517},[507,68770,68771],{"class":1382},"rng_",[507,68773,68774],{"class":517},": np.random.Generator) -> np.ndarray:\n",[507,68776,68777],{"class":509,"line":2442},[507,68778,1468],{"class":730},[507,68780,68781],{"class":509,"line":2447},[507,68782,68783],{"class":730},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[507,68785,68786],{"class":509,"line":2463},[507,68787,1468],{"class":730},[507,68789,68790,68792,68795,68798,68800,68802,68805,68808,68810,68813,68815,68817,68819],{"class":509,"line":2468},[507,68791,2504],{"class":513},[507,68793,68794],{"class":517}," rng_.",[507,68796,68797],{"class":576},"binomial",[507,68799,580],{"class":517},[507,68801,625],{"class":583},[507,68803,68804],{"class":517},", p, ",[507,68806,68807],{"class":2155},"size",[507,68809,573],{"class":572},[507,68811,68812],{"class":517},"shots).",[507,68814,66759],{"class":576},[507,68816,580],{"class":517},[507,68818,1420],{"class":572},[507,68820,587],{"class":517},[507,68822,68823],{"class":509,"line":2488},[507,68824,556],{"emptyLinePlaceholder":133},[507,68826,68827],{"class":509,"line":2501},[507,68828,556],{"emptyLinePlaceholder":133},[507,68830,68831,68833,68836,68838,68841,68844,68847,68849,68852,68855,68857],{"class":509,"line":59352},[507,68832,1370],{"class":513},[507,68834,68835],{"class":576}," summarize_model",[507,68837,580],{"class":517},[507,68839,68840],{"class":1382},"model",[507,68842,68843],{"class":517},": QubitModel, ",[507,68845,68846],{"class":1382},"bits",[507,68848,64107],{"class":517},[507,68850,68851],{"class":1382},"epsilons",[507,68853,68854],{"class":517},": List[",[507,68856,1406],{"class":572},[507,68858,68859],{"class":517},"]) -> Dict:\n",[507,68861,68862],{"class":509,"line":59357},[507,68863,1468],{"class":730},[507,68865,68866],{"class":509,"line":59364},[507,68867,68868],{"class":730},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[507,68870,68871],{"class":509,"line":59378},[507,68872,1468],{"class":730},[507,68874,68875,68877,68879,68881],{"class":509,"line":59383},[507,68876,68277],{"class":517},[507,68878,573],{"class":572},[507,68880,2095],{"class":572},[507,68882,68883],{"class":517},"(bits.size)\n",[507,68885,68886,68889,68891,68893,68896,68899],{"class":509,"line":59396},[507,68887,68888],{"class":517},"    ones ",[507,68890,573],{"class":572},[507,68892,2095],{"class":572},[507,68894,68895],{"class":517},"(bits.",[507,68897,68898],{"class":576},"sum",[507,68900,22087],{"class":517},[507,68902,68903,68906,68908,68910,68913,68915],{"class":509,"line":59412},[507,68904,68905],{"class":517},"    zeros ",[507,68907,573],{"class":572},[507,68909,2095],{"class":572},[507,68911,68912],{"class":517},"(n ",[507,68914,2367],{"class":572},[507,68916,68917],{"class":517}," ones)\n",[507,68919,68920,68923,68925,68928,68930,68932,68934,68936,68938,68940,68942,68944,68946,68949],{"class":509,"line":59432},[507,68921,68922],{"class":517},"    frac ",[507,68924,573],{"class":572},[507,68926,68927],{"class":517}," ones ",[507,68929,645],{"class":572},[507,68931,67941],{"class":517},[507,68933,1645],{"class":513},[507,68935,67941],{"class":517},[507,68937,1651],{"class":572},[507,68939,21258],{"class":583},[507,68941,8480],{"class":513},[507,68943,59620],{"class":572},[507,68945,580],{"class":517},[507,68947,68948],{"class":730},"\"nan\"",[507,68950,587],{"class":517},[507,68952,68953,68956,68958,68960],{"class":509,"line":59440},[507,68954,68955],{"class":517},"    lo, hi ",[507,68957,573],{"class":572},[507,68959,67876],{"class":576},[507,68961,68962],{"class":517},"(ones, n)\n",[507,68964,68965],{"class":509,"line":59446},[507,68966,556],{"emptyLinePlaceholder":133},[507,68968,68969,68972,68974],{"class":509,"line":59451},[507,68970,68971],{"class":517},"    row ",[507,68973,573],{"class":572},[507,68975,23723],{"class":517},[507,68977,68978,68981],{"class":509,"line":59462},[507,68979,68980],{"class":730},"        \"Model\"",[507,68982,68983],{"class":517},": model.name,\n",[507,68985,68986,68989],{"class":509,"line":59468},[507,68987,68988],{"class":730},"        \"Description\"",[507,68990,68991],{"class":517},": model.description,\n",[507,68993,68994,68997],{"class":509,"line":59480},[507,68995,68996],{"class":730},"        \"Configured p (target)\"",[507,68998,68999],{"class":517},": model.satisfaction_prob,\n",[507,69001,69002,69005],{"class":509,"line":59486},[507,69003,69004],{"class":730},"        \"Shots\"",[507,69006,69007],{"class":517},": n,\n",[507,69009,69010,69013],{"class":509,"line":59491},[507,69011,69012],{"class":730},"        \"Ones (successes)\"",[507,69014,69015],{"class":517},": ones,\n",[507,69017,69018,69021],{"class":509,"line":59496},[507,69019,69020],{"class":730},"        \"Zeros (failures)\"",[507,69022,69023],{"class":517},": zeros,\n",[507,69025,69026,69029],{"class":509,"line":59501},[507,69027,69028],{"class":730},"        \"Ones fraction (observed)\"",[507,69030,69031],{"class":517},": frac,\n",[507,69033,69034,69037,69040,69043],{"class":509,"line":59524},[507,69035,69036],{"class":730},"        \"Wilson CI 95",[507,69038,69039],{"class":583},"% lo",[507,69041,69042],{"class":730},"w\"",[507,69044,69045],{"class":517},": lo,\n",[507,69047,69048,69050,69053,69056],{"class":509,"line":59530},[507,69049,69036],{"class":730},[507,69051,69052],{"class":583},"% hi",[507,69054,69055],{"class":730},"gh\"",[507,69057,69058],{"class":517},": hi,\n",[507,69060,69061,69064,69067,69069],{"class":509,"line":59550},[507,69062,69063],{"class":730},"        \"CI width (95%)\"",[507,69065,69066],{"class":517},": hi ",[507,69068,2367],{"class":572},[507,69070,69071],{"class":517}," lo,\n",[507,69073,69074],{"class":509,"line":59555},[507,69075,59483],{"class":517},[507,69077,69078,69080,69083,69085],{"class":509,"line":59560},[507,69079,1916],{"class":513},[507,69081,69082],{"class":517}," eps ",[507,69084,1636],{"class":513},[507,69086,69087],{"class":517}," epsilons:\n",[507,69089,69090,69093,69095,69098,69100,69102,69105,69107,69109,69111,69113,69116,69118,69120,69122],{"class":509,"line":59570},[507,69091,69092],{"class":517},"        row[",[507,69094,22278],{"class":513},[507,69096,69097],{"class":730},"\"Required shots for ±",[507,69099,2810],{"class":583},[507,69101,1420],{"class":572},[507,69103,69104],{"class":517},"(eps",[507,69106,2391],{"class":572},[507,69108,5682],{"class":583},[507,69110,3649],{"class":517},[507,69112,2872],{"class":583},[507,69114,69115],{"class":730},"%\"",[507,69117,8206],{"class":517},[507,69119,573],{"class":572},[507,69121,68150],{"class":576},[507,69123,69124],{"class":517},"(model.satisfaction_prob, eps)\n",[507,69126,69127,69129],{"class":509,"line":59582},[507,69128,2504],{"class":513},[507,69130,69131],{"class":517}," row\n",[507,69133,69134],{"class":509,"line":59594},[507,69135,556],{"emptyLinePlaceholder":133},[507,69137,69138],{"class":509,"line":59606},[507,69139,556],{"emptyLinePlaceholder":133},[507,69141,69142,69144,69147,69149,69151,69153,69155,69157,69160,69162,69164,69166,69169,69171,69173,69175,69177],{"class":509,"line":59626},[507,69143,1370],{"class":513},[507,69145,69146],{"class":576}," apply_costs",[507,69148,580],{"class":517},[507,69150,68762],{"class":1382},[507,69152,1403],{"class":517},[507,69154,1420],{"class":572},[507,69156,622],{"class":517},[507,69158,69159],{"class":1382},"cost_per_shot",[507,69161,1403],{"class":517},[507,69163,1406],{"class":572},[507,69165,622],{"class":517},[507,69167,69168],{"class":1382},"fixed_overhead",[507,69170,1403],{"class":517},[507,69172,1406],{"class":572},[507,69174,58900],{"class":517},[507,69176,1406],{"class":572},[507,69178,1728],{"class":517},[507,69180,69181],{"class":509,"line":59632},[507,69182,1468],{"class":730},[507,69184,69185],{"class":509,"line":59656},[507,69186,69187],{"class":730},"    Calculate illustrative cost based on shots and cost model.\n",[507,69189,69190],{"class":509,"line":59679},[507,69191,1468],{"class":730},[507,69193,69194,69196,69199,69201,69204,69206],{"class":509,"line":59695},[507,69195,2504],{"class":513},[507,69197,69198],{"class":517}," fixed_overhead ",[507,69200,2107],{"class":572},[507,69202,69203],{"class":517}," shots ",[507,69205,2391],{"class":572},[507,69207,69208],{"class":517}," cost_per_shot\n",[507,69210,69211],{"class":509,"line":59700},[507,69212,556],{"emptyLinePlaceholder":133},[507,69214,69215],{"class":509,"line":59727},[507,69216,556],{"emptyLinePlaceholder":133},[507,69218,69219],{"class":509,"line":59732},[507,69220,69221],{"class":562},"# ---------- Run the simulation ----------\n",[507,69223,69224],{"class":509,"line":59756},[507,69225,556],{"emptyLinePlaceholder":133},[507,69227,69228,69231,69233,69236,69238],{"class":509,"line":59778},[507,69229,69230],{"class":517},"all_bitstrings: Dict[",[507,69232,58897],{"class":572},[507,69234,69235],{"class":517},", np.ndarray] ",[507,69237,573],{"class":572},[507,69239,23708],{"class":517},[507,69241,69242,69245,69247],{"class":509,"line":59799},[507,69243,69244],{"class":517},"summary_rows: List[Dict] ",[507,69246,573],{"class":572},[507,69248,1910],{"class":517},[507,69250,69251],{"class":509,"line":59804},[507,69252,556],{"emptyLinePlaceholder":133},[507,69254,69255,69257,69259,69262],{"class":509,"line":59818},[507,69256,8525],{"class":572},[507,69258,580],{"class":517},[507,69260,69261],{"class":730},"\"Running simulation...\"",[507,69263,587],{"class":517},[507,69265,69266,69268,69271,69273,69276],{"class":509,"line":59833},[507,69267,1630],{"class":513},[507,69269,69270],{"class":517}," m ",[507,69272,1636],{"class":513},[507,69274,69275],{"class":583}," DEFAULT_MODELS",[507,69277,1728],{"class":517},[507,69279,69280,69283,69285,69287,69290,69292,69294,69297],{"class":509,"line":59851},[507,69281,69282],{"class":517},"    bits ",[507,69284,573],{"class":572},[507,69286,68749],{"class":576},[507,69288,69289],{"class":517},"(m.satisfaction_prob, ",[507,69291,68581],{"class":583},[507,69293,12248],{"class":517},[507,69295,69296],{"class":730},"\"shots_per_model\"",[507,69298,69299],{"class":517},"], rng)\n",[507,69301,69302,69305,69307],{"class":509,"line":59856},[507,69303,69304],{"class":517},"    all_bitstrings[m.name] ",[507,69306,573],{"class":572},[507,69308,69309],{"class":517}," bits\n",[507,69311,69312,69315,69317,69319,69322,69325,69327,69329,69332],{"class":509,"line":59861},[507,69313,69314],{"class":517},"    summary_rows.",[507,69316,1939],{"class":576},[507,69318,580],{"class":517},[507,69320,69321],{"class":576},"summarize_model",[507,69323,69324],{"class":517},"(m, bits, ",[507,69326,68581],{"class":583},[507,69328,12248],{"class":517},[507,69330,69331],{"class":730},"\"epsilons\"",[507,69333,69334],{"class":517},"]))\n",[507,69336,69337],{"class":509,"line":59876},[507,69338,556],{"emptyLinePlaceholder":133},[507,69340,69341,69344,69346,69348,69350],{"class":509,"line":59882},[507,69342,69343],{"class":517},"summary_df ",[507,69345,573],{"class":572},[507,69347,58042],{"class":517},[507,69349,58045],{"class":576},[507,69351,69352],{"class":517},"(summary_rows)\n",[507,69354,69355],{"class":509,"line":59888},[507,69356,556],{"emptyLinePlaceholder":133},[507,69358,69359],{"class":509,"line":59894},[507,69360,69361],{"class":562},"# Add cost estimates\n",[507,69363,69364,69367,69369,69372,69374,69377,69379,69382],{"class":509,"line":59899},[507,69365,69366],{"class":517},"cps ",[507,69368,573],{"class":572},[507,69370,69371],{"class":583}," CONFIG",[507,69373,12248],{"class":517},[507,69375,69376],{"class":730},"\"cost_model\"",[507,69378,1755],{"class":517},[507,69380,69381],{"class":730},"\"default_cost_per_shot_usd\"",[507,69383,1794],{"class":517},[507,69385,69386,69389,69391,69393,69395,69397,69399,69402],{"class":509,"line":59904},[507,69387,69388],{"class":517},"over ",[507,69390,573],{"class":572},[507,69392,69371],{"class":583},[507,69394,12248],{"class":517},[507,69396,69376],{"class":730},[507,69398,1755],{"class":517},[507,69400,69401],{"class":730},"\"fixed_overhead_usd\"",[507,69403,1794],{"class":517},[507,69405,69406,69408,69411,69413,69415,69417,69420,69422,69424],{"class":509,"line":59936},[507,69407,1645],{"class":513},[507,69409,69410],{"class":517}," cps ",[507,69412,1651],{"class":572},[507,69414,21258],{"class":583},[507,69416,22167],{"class":513},[507,69418,69419],{"class":517}," over ",[507,69421,1651],{"class":572},[507,69423,21258],{"class":583},[507,69425,1728],{"class":517},[507,69427,69428,69431,69434,69436,69438],{"class":509,"line":59963},[507,69429,69430],{"class":517},"    summary_df[",[507,69432,69433],{"class":730},"\"Illustrative cost (USD)\"",[507,69435,8206],{"class":517},[507,69437,573],{"class":572},[507,69439,2177],{"class":517},[507,69441,69442,69445,69447,69449,69452,69455],{"class":509,"line":59989},[507,69443,69444],{"class":576},"        apply_costs",[507,69446,580],{"class":517},[507,69448,1420],{"class":572},[507,69450,69451],{"class":517},"(row[",[507,69453,69454],{"class":730},"\"Shots\"",[507,69456,69457],{"class":517},"]), cps, over)\n",[507,69459,69460,69462,69465,69467,69470,69473],{"class":509,"line":59994},[507,69461,2267],{"class":513},[507,69463,69464],{"class":517}," _, row ",[507,69466,1636],{"class":513},[507,69468,69469],{"class":517}," summary_df.",[507,69471,69472],{"class":576},"iterrows",[507,69474,781],{"class":517},[507,69476,69477],{"class":509,"line":60008},[507,69478,59040],{"class":517},[507,69480,69481],{"class":509,"line":60023},[507,69482,556],{"emptyLinePlaceholder":133},[507,69484,69485],{"class":509,"line":60028},[507,69486,69487],{"class":562},"# ---------- Save artifacts ----------\n",[507,69489,69490],{"class":509,"line":60033},[507,69491,556],{"emptyLinePlaceholder":133},[507,69493,69494],{"class":509,"line":60047},[507,69495,69496],{"class":562},"# Save raw bitstrings and summary CSV\n",[507,69498,69499,69501,69504,69506,69509,69511],{"class":509,"line":60052},[507,69500,1630],{"class":513},[507,69502,69503],{"class":517}," name, arr ",[507,69505,1636],{"class":513},[507,69507,69508],{"class":517}," all_bitstrings.",[507,69510,22607],{"class":576},[507,69512,1930],{"class":517},[507,69514,69515,69518,69520,69522,69525,69527,69530,69533,69535,69538,69540,69543,69545,69547],{"class":509,"line":60058},[507,69516,69517],{"class":517},"    out_path ",[507,69519,573],{"class":572},[507,69521,64432],{"class":513},[507,69523,69524],{"class":730},"\"bitstrings_",[507,69526,2810],{"class":583},[507,69528,69529],{"class":517},"name.",[507,69531,69532],{"class":576},"replace",[507,69534,580],{"class":517},[507,69536,69537],{"class":730},"' '",[507,69539,622],{"class":517},[507,69541,69542],{"class":730},"'_'",[507,69544,3649],{"class":517},[507,69546,2872],{"class":583},[507,69548,69549],{"class":730},".csv\"\n",[507,69551,69552,69555,69557,69560,69563,69566,69569,69572,69575,69577,69579],{"class":509,"line":60064},[507,69553,69554],{"class":517},"    pd.",[507,69556,58045],{"class":576},[507,69558,69559],{"class":517},"({",[507,69561,69562],{"class":730},"\"bit\"",[507,69564,69565],{"class":517},": arr}).",[507,69567,69568],{"class":576},"to_csv",[507,69570,69571],{"class":517},"(out_path, ",[507,69573,69574],{"class":2155},"index",[507,69576,573],{"class":572},[507,69578,21964],{"class":583},[507,69580,587],{"class":517},[507,69582,69583],{"class":509,"line":60070},[507,69584,556],{"emptyLinePlaceholder":133},[507,69586,69587,69590,69592],{"class":509,"line":60075},[507,69588,69589],{"class":517},"summary_csv_path ",[507,69591,573],{"class":572},[507,69593,69594],{"class":730}," \"summary.csv\"\n",[507,69596,69597,69600,69602,69605,69607,69609,69611],{"class":509,"line":60080},[507,69598,69599],{"class":517},"summary_df.",[507,69601,69568],{"class":576},[507,69603,69604],{"class":517},"(summary_csv_path, ",[507,69606,69574],{"class":2155},[507,69608,573],{"class":572},[507,69610,21964],{"class":583},[507,69612,587],{"class":517},[507,69614,69615],{"class":509,"line":60085},[507,69616,556],{"emptyLinePlaceholder":133},[507,69618,69619],{"class":509,"line":60099},[507,69620,69621],{"class":562},"# Save config JSON\n",[507,69623,69624,69627,69630,69632,69635,69637,69640,69642,69644],{"class":509,"line":60104},[507,69625,69626],{"class":513},"with",[507,69628,69629],{"class":572}," open",[507,69631,580],{"class":517},[507,69633,69634],{"class":730},"\"simulation_config.json\"",[507,69636,622],{"class":517},[507,69638,69639],{"class":730},"\"w\"",[507,69641,655],{"class":517},[507,69643,521],{"class":513},[507,69645,69646],{"class":517}," f:\n",[507,69648,69649,69652,69655,69657,69659,69662,69665,69667,69669],{"class":509,"line":60115},[507,69650,69651],{"class":517},"    json.",[507,69653,69654],{"class":576},"dump",[507,69656,580],{"class":517},[507,69658,68581],{"class":583},[507,69660,69661],{"class":517},", f, ",[507,69663,69664],{"class":2155},"indent",[507,69666,573],{"class":572},[507,69668,584],{"class":583},[507,69670,587],{"class":517},[507,69672,69673],{"class":509,"line":60123},[507,69674,556],{"emptyLinePlaceholder":133},[507,69676,69677,69679,69681,69684],{"class":509,"line":60133},[507,69678,8525],{"class":572},[507,69680,580],{"class":517},[507,69682,69683],{"class":730},"\"Saved data to CSV and JSON files.\"",[507,69685,587],{"class":517},[507,69687,69688],{"class":509,"line":60138},[507,69689,556],{"emptyLinePlaceholder":133},[507,69691,69692],{"class":509,"line":60143},[507,69693,69694],{"class":562},"# ---------- Visualizations ----------\n",[507,69696,69697],{"class":509,"line":60148},[507,69698,556],{"emptyLinePlaceholder":133},[507,69700,69701],{"class":509,"line":60155},[507,69702,69703],{"class":562},"# Figure 1: Observed ones fraction with Wilson CI\n",[507,69705,69706,69709,69712,69714,69717],{"class":509,"line":60171},[507,69707,69708],{"class":517},"plt.style.",[507,69710,69711],{"class":576},"use",[507,69713,580],{"class":517},[507,69715,69716],{"class":730},"'seaborn-v0_8-whitegrid'",[507,69718,587],{"class":517},[507,69720,69721,69723,69725,69727,69729,69731,69733,69735,69737,69739],{"class":509,"line":60180},[507,69722,25376],{"class":517},[507,69724,61616],{"class":576},[507,69726,580],{"class":517},[507,69728,58233],{"class":2155},[507,69730,573],{"class":572},[507,69732,580],{"class":517},[507,69734,23805],{"class":583},[507,69736,622],{"class":517},[507,69738,63712],{"class":583},[507,69740,22540],{"class":517},[507,69742,69743,69746,69748,69750,69753,69755,69757],{"class":509,"line":60189},[507,69744,69745],{"class":517},"x ",[507,69747,573],{"class":572},[507,69749,1616],{"class":517},[507,69751,69752],{"class":576},"arange",[507,69754,580],{"class":517},[507,69756,1763],{"class":572},[507,69758,69759],{"class":517},"(summary_df))\n",[507,69761,69762,69765,69767,69770,69773,69775,69778],{"class":509,"line":60198},[507,69763,69764],{"class":517},"y ",[507,69766,573],{"class":572},[507,69768,69769],{"class":517}," summary_df[",[507,69771,69772],{"class":730},"\"Ones fraction (observed)\"",[507,69774,22905],{"class":517},[507,69776,69777],{"class":576},"to_numpy",[507,69779,781],{"class":517},[507,69781,69782],{"class":509,"line":60207},[507,69783,556],{"emptyLinePlaceholder":133},[507,69785,69786],{"class":509,"line":60230},[507,69787,69788],{"class":562},"# --- FIX APPLIED HERE ---\n",[507,69790,69791],{"class":509,"line":60236},[507,69792,69793],{"class":562},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[507,69795,69796],{"class":509,"line":60241},[507,69797,69798],{"class":562},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[507,69800,69801,69804,69806,69809,69811,69813,69816,69818,69820,69822,69824],{"class":509,"line":60246},[507,69802,69803],{"class":517},"lower_error ",[507,69805,573],{"class":572},[507,69807,69808],{"class":517}," y ",[507,69810,2367],{"class":572},[507,69812,69769],{"class":517},[507,69814,69815],{"class":730},"\"Wilson CI 95",[507,69817,69039],{"class":583},[507,69819,69042],{"class":730},[507,69821,22905],{"class":517},[507,69823,69777],{"class":576},[507,69825,781],{"class":517},[507,69827,69828,69831,69833,69835,69837,69839,69841,69843,69845,69847,69849],{"class":509,"line":60251},[507,69829,69830],{"class":517},"upper_error ",[507,69832,573],{"class":572},[507,69834,69769],{"class":517},[507,69836,69815],{"class":730},[507,69838,69052],{"class":583},[507,69840,69055],{"class":730},[507,69842,22905],{"class":517},[507,69844,69777],{"class":576},[507,69846,1677],{"class":517},[507,69848,2367],{"class":572},[507,69850,69851],{"class":517}," y\n",[507,69853,69854,69857,69859,69861,69863,69866,69868,69871,69873],{"class":509,"line":60261},[507,69855,69856],{"class":517},"yerr ",[507,69858,573],{"class":572},[507,69860,1616],{"class":517},[507,69862,23073],{"class":576},[507,69864,69865],{"class":517},"([np.",[507,69867,64259],{"class":576},[507,69869,69870],{"class":517},"(lower_error), np.",[507,69872,64259],{"class":576},[507,69874,69875],{"class":517},"(upper_error)])\n",[507,69877,69878],{"class":509,"line":60270},[507,69879,69880],{"class":562},"# --- END OF FIX ---\n",[507,69882,69883],{"class":509,"line":60279},[507,69884,556],{"emptyLinePlaceholder":133},[507,69886,69887,69889,69892,69895,69897,69899,69902,69904,69907,69909,69912],{"class":509,"line":60284},[507,69888,25376],{"class":517},[507,69890,69891],{"class":576},"bar",[507,69893,69894],{"class":517},"(x, y, ",[507,69896,64350],{"class":2155},[507,69898,573],{"class":572},[507,69900,69901],{"class":730},"'skyblue'",[507,69903,622],{"class":517},[507,69905,69906],{"class":2155},"edgecolor",[507,69908,573],{"class":572},[507,69910,69911],{"class":730},"'black'",[507,69913,587],{"class":517},[507,69915,69916,69918,69921,69923,69926,69928,69931,69934,69936,69939,69941,69944,69946,69948,69950,69952,69954,69957,69959,69962,69964,69966],{"class":509,"line":60290},[507,69917,25376],{"class":517},[507,69919,69920],{"class":576},"errorbar",[507,69922,69894],{"class":517},[507,69924,69925],{"class":2155},"yerr",[507,69927,573],{"class":572},[507,69929,69930],{"class":517},"yerr, ",[507,69932,69933],{"class":2155},"fmt",[507,69935,573],{"class":572},[507,69937,69938],{"class":730},"\"none\"",[507,69940,622],{"class":517},[507,69942,69943],{"class":2155},"capsize",[507,69945,573],{"class":572},[507,69947,58245],{"class":583},[507,69949,622],{"class":517},[507,69951,64350],{"class":2155},[507,69953,573],{"class":572},[507,69955,69956],{"class":730},"\"black\"",[507,69958,622],{"class":517},[507,69960,69961],{"class":2155},"elinewidth",[507,69963,573],{"class":572},[507,69965,66210],{"class":583},[507,69967,587],{"class":517},[507,69969,69970,69972,69974,69977,69980,69982,69984,69986,69989,69991,69993,69995,69998,70000,70003],{"class":509,"line":60306},[507,69971,25376],{"class":517},[507,69973,25521],{"class":576},[507,69975,69976],{"class":517},"(x, summary_df[",[507,69978,69979],{"class":730},"\"Model\"",[507,69981,22905],{"class":517},[507,69983,24370],{"class":576},[507,69985,13950],{"class":517},[507,69987,69988],{"class":2155},"rotation",[507,69990,573],{"class":572},[507,69992,48588],{"class":583},[507,69994,622],{"class":517},[507,69996,69997],{"class":2155},"ha",[507,69999,573],{"class":572},[507,70001,70002],{"class":730},"\"right\"",[507,70004,587],{"class":517},[507,70006,70007,70009,70011,70013,70016],{"class":509,"line":60317},[507,70008,25376],{"class":517},[507,70010,25581],{"class":576},[507,70012,580],{"class":517},[507,70014,70015],{"class":730},"\"Observed Ones Fraction\"",[507,70017,587],{"class":517},[507,70019,70020,70022,70024,70026,70029,70031,70034,70036,70039],{"class":509,"line":60332},[507,70021,25376],{"class":517},[507,70023,25595],{"class":576},[507,70025,580],{"class":517},[507,70027,70028],{"class":730},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[507,70030,622],{"class":517},[507,70032,70033],{"class":2155},"fontsize",[507,70035,573],{"class":572},[507,70037,70038],{"class":583},"14",[507,70040,587],{"class":517},[507,70042,70043,70045,70047],{"class":509,"line":60342},[507,70044,25376],{"class":517},[507,70046,58370],{"class":576},[507,70048,781],{"class":517},[507,70050,70051,70054,70056],{"class":509,"line":60349},[507,70052,70053],{"class":517},"fig1_path ",[507,70055,573],{"class":572},[507,70057,70058],{"class":730}," \"fig_ones_fraction.png\"\n",[507,70060,70061,70063,70066,70069,70071,70073,70075,70077,70080,70082,70085],{"class":509,"line":60358},[507,70062,25376],{"class":517},[507,70064,70065],{"class":576},"savefig",[507,70067,70068],{"class":517},"(fig1_path, ",[507,70070,63717],{"class":2155},[507,70072,573],{"class":572},[507,70074,63722],{"class":583},[507,70076,622],{"class":517},[507,70078,70079],{"class":2155},"bbox_inches",[507,70081,573],{"class":572},[507,70083,70084],{"class":730},"'tight'",[507,70086,587],{"class":517},[507,70088,70089,70091,70093],{"class":509,"line":60363},[507,70090,25376],{"class":517},[507,70092,25613],{"class":576},[507,70094,781],{"class":517},[507,70096,70097],{"class":509,"line":60385},[507,70098,556],{"emptyLinePlaceholder":133},[507,70100,70101],{"class":509,"line":60390},[507,70102,70103],{"class":562},"# Figure 2: Required shots at several ±epsilon targets\n",[507,70105,70106,70108,70110,70112,70114,70116,70118,70120,70122,70124],{"class":509,"line":60407},[507,70107,25376],{"class":517},[507,70109,61616],{"class":576},[507,70111,580],{"class":517},[507,70113,58233],{"class":2155},[507,70115,573],{"class":572},[507,70117,580],{"class":517},[507,70119,23805],{"class":583},[507,70121,622],{"class":517},[507,70123,63712],{"class":583},[507,70125,22540],{"class":517},[507,70127,70128,70131,70133],{"class":509,"line":60424},[507,70129,70130],{"class":517},"width ",[507,70132,573],{"class":572},[507,70134,70135],{"class":583}," 0.2\n",[507,70137,70138,70140,70142,70144,70146,70148,70150,70152,70154],{"class":509,"line":60429},[507,70139,69745],{"class":517},[507,70141,573],{"class":572},[507,70143,1616],{"class":517},[507,70145,69752],{"class":576},[507,70147,580],{"class":517},[507,70149,1763],{"class":572},[507,70151,580],{"class":517},[507,70153,68406],{"class":583},[507,70155,22540],{"class":517},[507,70157,70158,70161,70163,70165,70168,70170,70173,70175,70178],{"class":509,"line":60442},[507,70159,70160],{"class":517},"colors ",[507,70162,573],{"class":572},[507,70164,8427],{"class":517},[507,70166,70167],{"class":730},"'#ff9999'",[507,70169,2819],{"class":517},[507,70171,70172],{"class":730},"'#66b3ff'",[507,70174,2819],{"class":517},[507,70176,70177],{"class":730},"'#99ff99'",[507,70179,1794],{"class":517},[507,70181,70182],{"class":509,"line":60456},[507,70183,556],{"emptyLinePlaceholder":133},[507,70185,70186,70188,70191,70193,70195,70197,70199,70201,70203],{"class":509,"line":60461},[507,70187,1630],{"class":513},[507,70189,70190],{"class":517}," i, eps ",[507,70192,1636],{"class":513},[507,70194,1957],{"class":572},[507,70196,580],{"class":517},[507,70198,68581],{"class":583},[507,70200,12248],{"class":517},[507,70202,69331],{"class":730},[507,70204,70205],{"class":517},"]):\n",[507,70207,70208,70211,70213,70215,70218,70221,70223,70225,70227,70229],{"class":509,"line":60469},[507,70209,70210],{"class":517},"    req ",[507,70212,573],{"class":572},[507,70214,8427],{"class":517},[507,70216,70217],{"class":576},"shots_for_epsilon",[507,70219,70220],{"class":517},"(m.satisfaction_prob, eps) ",[507,70222,1630],{"class":513},[507,70224,69270],{"class":517},[507,70226,1636],{"class":513},[507,70228,69275],{"class":583},[507,70230,1794],{"class":517},[507,70232,70233,70236,70238,70241,70243,70245,70247,70250,70252,70254,70256,70259,70261,70264,70267,70269,70271,70273,70275,70277,70280,70282,70284,70286],{"class":509,"line":60474},[507,70234,70235],{"class":517},"    plt.",[507,70237,69891],{"class":576},[507,70239,70240],{"class":517},"(x ",[507,70242,2107],{"class":572},[507,70244,8246],{"class":517},[507,70246,2391],{"class":572},[507,70248,70249],{"class":517}," width, req, width, 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±ε\"",[507,70356,622],{"class":517},[507,70358,70033],{"class":2155},[507,70360,573],{"class":572},[507,70362,70038],{"class":583},[507,70364,587],{"class":517},[507,70366,70367,70369,70372,70374,70377],{"class":509,"line":60516},[507,70368,25376],{"class":517},[507,70370,70371],{"class":576},"yscale",[507,70373,580],{"class":517},[507,70375,70376],{"class":730},"'log'",[507,70378,587],{"class":517},[507,70380,70381,70383,70385,70387,70389,70391,70394],{"class":509,"line":60527},[507,70382,25376],{"class":517},[507,70384,25558],{"class":576},[507,70386,580],{"class":517},[507,70388,25595],{"class":2155},[507,70390,573],{"class":572},[507,70392,70393],{"class":730},"\"Error Tolerance\"",[507,70395,587],{"class":517},[507,70397,70398,70400,70402],{"class":509,"line":60536},[507,70399,25376],{"class":517},[507,70401,58370],{"class":576},[507,70403,781],{"class":517},[507,70405,70406,70409,70411],{"class":509,"line":60542},[507,70407,70408],{"class":517},"fig2_path ",[507,70410,573],{"class":572},[507,70412,70413],{"class":730}," \"fig_required_shots.png\"\n",[507,70415,70416,70418,70420,70423,70425,70427,70429,70431,70433,70435,70437],{"class":509,"line":60552},[507,70417,25376],{"class":517},[507,70419,70065],{"class":576},[507,70421,70422],{"class":517},"(fig2_path, ",[507,70424,63717],{"class":2155},[507,70426,573],{"class":572},[507,70428,63722],{"class":583},[507,70430,622],{"class":517},[507,70432,70079],{"class":2155},[507,70434,573],{"class":572},[507,70436,70084],{"class":730},[507,70438,587],{"class":517},[507,70440,70441,70443,70445],{"class":509,"line":60562},[507,70442,25376],{"class":517},[507,70444,25613],{"class":576},[507,70446,781],{"class":517},[507,70448,70449],{"class":509,"line":60571},[507,70450,556],{"emptyLinePlaceholder":133},[507,70452,70453,70455,70457,70460],{"class":509,"line":60580},[507,70454,8525],{"class":572},[507,70456,580],{"class":517},[507,70458,70459],{"class":730},"\"Generated and saved plots.\"",[507,70461,587],{"class":517},[507,70463,70464],{"class":509,"line":60591},[507,70465,556],{"emptyLinePlaceholder":133},[507,70467,70468],{"class":509,"line":60603},[507,70469,70470],{"class":562},"# ---------- Display interactive table ----------\n",[507,70472,70473],{"class":509,"line":60614},[507,70474,556],{"emptyLinePlaceholder":133},[507,70476,70477,70480,70482,70485,70487,70490],{"class":509,"line":60619},[507,70478,70479],{"class":576},"display",[507,70481,580],{"class":517},[507,70483,70484],{"class":576},"Markdown",[507,70486,580],{"class":517},[507,70488,70489],{"class":730},"\"##  Qubit Sampling Budget - Summary\"",[507,70491,22540],{"class":517},[507,70493,70494,70496,70499,70502],{"class":509,"line":60624},[507,70495,70479],{"class":576},[507,70497,70498],{"class":517},"(summary_df.style.",[507,70500,70501],{"class":576},"format",[507,70503,70504],{"class":517},"({\n",[507,70506,70507,70510,70512,70514,70516,70519,70521,70523],{"class":509,"line":60629},[507,70508,70509],{"class":730},"    \"Configured p (target)\"",[507,70511,1403],{"class":517},[507,70513,22281],{"class":730},[507,70515,2810],{"class":583},[507,70517,70518],{"class":513},":.2f",[507,70520,2872],{"class":583},[507,70522,22281],{"class":730},[507,70524,1409],{"class":517},[507,70526,70527,70530,70532,70534,70536,70539,70541,70543],{"class":509,"line":60639},[507,70528,70529],{"class":730},"    \"Ones fraction (observed)\"",[507,70531,1403],{"class":517},[507,70533,22281],{"class":730},[507,70535,2810],{"class":583},[507,70537,70538],{"class":513},":.4f",[507,70540,2872],{"class":583},[507,70542,22281],{"class":730},[507,70544,1409],{"class":517},[507,70546,70547,70550,70552,70554,70556,70558,70560,70562,70564,70566],{"class":509,"line":60646},[507,70548,70549],{"class":730},"    \"Wilson CI 95",[507,70551,69039],{"class":583},[507,70553,69042],{"class":730},[507,70555,1403],{"class":517},[507,70557,22281],{"class":730},[507,70559,2810],{"class":583},[507,70561,70538],{"class":513},[507,70563,2872],{"class":583},[507,70565,22281],{"class":730},[507,70567,1409],{"class":517},[507,70569,70570,70572,70574,70576,70578,70580,70582,70584,70586,70588],{"class":509,"line":60654},[507,70571,70549],{"class":730},[507,70573,69052],{"class":583},[507,70575,69055],{"class":730},[507,70577,1403],{"class":517},[507,70579,22281],{"class":730},[507,70581,2810],{"class":583},[507,70583,70538],{"class":513},[507,70585,2872],{"class":583},[507,70587,22281],{"class":730},[507,70589,1409],{"class":517},[507,70591,70592,70595,70597,70599,70601,70603,70605,70607],{"class":509,"line":60665},[507,70593,70594],{"class":730},"    \"CI width (95%)\"",[507,70596,1403],{"class":517},[507,70598,22281],{"class":730},[507,70600,2810],{"class":583},[507,70602,70538],{"class":513},[507,70604,2872],{"class":583},[507,70606,22281],{"class":730},[507,70608,1409],{"class":517},[507,70610,70611,70614,70616,70618,70620,70622,70624],{"class":509,"line":60676},[507,70612,70613],{"class":730},"    \"Illustrative cost (USD)\"",[507,70615,1403],{"class":517},[507,70617,61680],{"class":730},[507,70619,2810],{"class":583},[507,70621,70518],{"class":513},[507,70623,2872],{"class":583},[507,70625,61726],{"class":730},[507,70627,70628],{"class":509,"line":60685},[507,70629,70630],{"class":517},"}))\n",[507,70632,70633],{"class":509,"line":60691},[507,70634,556],{"emptyLinePlaceholder":133},[507,70636,70637],{"class":509,"line":60701},[507,70638,556],{"emptyLinePlaceholder":133},[507,70640,70641],{"class":509,"line":60711},[507,70642,70643],{"class":562},"# ---------- Build a ready-to-run Colab notebook ----------\n",[507,70645,70646],{"class":509,"line":60720},[507,70647,556],{"emptyLinePlaceholder":133},[507,70649,70650,70653,70655,70658],{"class":509,"line":60729},[507,70651,70652],{"class":517},"md_intro ",[507,70654,573],{"class":572},[507,70656,70657],{"class":513}," r",[507,70659,70660],{"class":58306},"\"\"\"\n",[507,70662,70663],{"class":509,"line":60738},[507,70664,70665],{"class":562},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[507,70667,70668],{"class":509,"line":60750},[507,70669,556],{"emptyLinePlaceholder":133},[507,70671,70672,70674,70677,70679,70682,70684,70686,70688,70691,70693,70696,70698],{"class":509,"line":60762},[507,70673,2377],{"class":583},[507,70675,70676],{"class":58306},"High school level ",[507,70678,580],{"class":583},[507,70680,70681],{"class":58306},"concise",[507,70683,3649],{"class":583},[507,70685,24985],{"class":58306},[507,70687,2377],{"class":583},[507,70689,70690],{"class":58306}," We treat each measurement ",[507,70692,580],{"class":583},[507,70694,70695],{"class":58306},"“single shot”",[507,70697,3649],{"class":583},[507,70699,70700],{"class":58306}," as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n",[507,70702,70703],{"class":509,"line":60771},[507,70704,556],{"emptyLinePlaceholder":133},[507,70706,70707,70709,70712,70714,70716,70718,70720,70722,70725,70727,70730,70732,70735,70737,70740,70742,70745,70747,70750,70752,70755,70758,70761,70764,70766,70768,70771,70773,70775,70778,70781,70783],{"class":509,"line":60776},[507,70708,2377],{"class":583},[507,70710,70711],{"class":58306},"Graduate level ",[507,70713,580],{"class":583},[507,70715,70681],{"class":58306},[507,70717,3649],{"class":583},[507,70719,24985],{"class":58306},[507,70721,2377],{"class":583},[507,70723,70724],{"class":58306}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[507,70726,580],{"class":583},[507,70728,70729],{"class":58306},"noisy intermediate-scale quantum",[507,70731,3649],{"class":583},[507,70733,70734],{"class":58306},", fault-tolerant ",[507,70736,580],{"class":583},[507,70738,70739],{"class":58306},"logical",[507,70741,3649],{"class":583},[507,70743,70744],{"class":58306},", topologically protected, and ideal. We report Wilson score intervals ",[507,70746,580],{"class":583},[507,70748,70749],{"class":58306},"95%",[507,70751,3649],{"class":583},[507,70753,70754],{"class":58306}," for observed proportions and estimate the required shot budget $n ",[507,70756,70757],{"class":572},"\\a",[507,70759,70760],{"class":58306},"pprox z^2",[507,70762,70763],{"class":572},"\\,",[507,70765,18],{"class":58306},[507,70767,580],{"class":583},[507,70769,70770],{"class":58306},"1-p",[507,70772,3649],{"class":583},[507,70774,645],{"class":58306},[507,70776,70777],{"class":572},"\\e",[507,70779,70780],{"class":58306},"psilon^2$ to bound absolute error by ±$",[507,70782,70777],{"class":572},[507,70784,70785],{"class":58306},"psilon$ at ~95% confidence.\n",[507,70787,70788],{"class":509,"line":60781},[507,70789,70660],{"class":58306},[507,70791,70792],{"class":509,"line":60786},[507,70793,556],{"emptyLinePlaceholder":133},[507,70795,70796,70799,70801,70803],{"class":509,"line":60796},[507,70797,70798],{"class":517},"code_config ",[507,70800,573],{"class":572},[507,70802,70657],{"class":513},[507,70804,70660],{"class":58306},[507,70806,70807],{"class":509,"line":60803},[507,70808,70809],{"class":562},"# @title Configuration\n",[507,70811,70812],{"class":509,"line":60810},[507,70813,70814],{"class":58306},"from dataclasses import dataclass\n",[507,70816,70817],{"class":509,"line":60821},[507,70818,70819],{"class":58306},"from typing import List, Dict\n",[507,70821,70822],{"class":509,"line":60832},[507,70823,70824],{"class":58306},"import numpy as np\n",[507,70826,70827],{"class":509,"line":60841},[507,70828,556],{"emptyLinePlaceholder":133},[507,70830,70831,70834],{"class":509,"line":60847},[507,70832,70833],{"class":58306},"RANDOM_SEED = 579345  ",[507,70835,70836],{"class":562},"# @param {type:\"number\"}\n",[507,70838,70839,70842],{"class":509,"line":60857},[507,70840,70841],{"class":58306},"SHOTS_PER_MODEL = 10000  ",[507,70843,70836],{"class":562},[507,70845,70846,70849,70852],{"class":509,"line":60865},[507,70847,70848],{"class":58306},"EPSILONS = ",[507,70850,70851],{"class":583},"[0.02, 0.01, 0.005]",[507,70853,70854],{"class":562},"  # @param\n",[507,70856,70857,70860],{"class":509,"line":60872},[507,70858,70859],{"class":58306},"COST_PER_SHOT_USD = 0.00005  ",[507,70861,70836],{"class":562},[507,70863,70864,70867],{"class":509,"line":60879},[507,70865,70866],{"class":58306},"FIXED_OVERHEAD_USD = 0.10    ",[507,70868,70836],{"class":562},[507,70870,70871],{"class":509,"line":60888},[507,70872,556],{"emptyLinePlaceholder":133},[507,70874,70875,70877,70879,70882],{"class":509,"line":60897},[507,70876,68346],{"class":58306},[507,70878,580],{"class":583},[507,70880,70881],{"class":58306},"frozen=True",[507,70883,587],{"class":583},[507,70885,70886],{"class":509,"line":60906},[507,70887,70888],{"class":58306},"class QubitModel:\n",[507,70890,70891],{"class":509,"line":60915},[507,70892,70893],{"class":58306},"    name: str\n",[507,70895,70896],{"class":509,"line":60926},[507,70897,70898],{"class":58306},"    satisfaction_prob: float\n",[507,70900,70901],{"class":509,"line":60938},[507,70902,70903],{"class":58306},"    description: str\n",[507,70905,70906],{"class":509,"line":60950},[507,70907,556],{"emptyLinePlaceholder":133},[507,70909,70910,70913,70916,70919],{"class":509,"line":60959},[507,70911,70912],{"class":58306},"MODELS: List",[507,70914,70915],{"class":583},"[QubitModel]",[507,70917,70918],{"class":58306}," = ",[507,70920,70921],{"class":583},"[\n",[507,70923,70924],{"class":509,"line":60964},[507,70925,70926],{"class":583},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[507,70928,70929],{"class":509,"line":60969},[507,70930,70931],{"class":583},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[507,70933,70934],{"class":509,"line":60974},[507,70935,70936],{"class":583},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[507,70938,70939],{"class":509,"line":60984},[507,70940,70941],{"class":583},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[507,70943,70944],{"class":509,"line":60991},[507,70945,70946],{"class":583},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[507,70948,70949],{"class":509,"line":60998},[507,70950,1794],{"class":583},[507,70952,70953],{"class":509,"line":61010},[507,70954,556],{"emptyLinePlaceholder":133},[507,70956,70957,70960,70962,70965],{"class":509,"line":61021},[507,70958,70959],{"class":58306},"rng = np.random.default_rng",[507,70961,580],{"class":583},[507,70963,70964],{"class":58306},"RANDOM_SEED",[507,70966,587],{"class":583},[507,70968,70969],{"class":509,"line":61030},[507,70970,70660],{"class":58306},[507,70972,70973],{"class":509,"line":61036},[507,70974,556],{"emptyLinePlaceholder":133},[507,70976,70977,70980,70982,70984],{"class":509,"line":61052},[507,70978,70979],{"class":517},"code_lib_and_run ",[507,70981,573],{"class":572},[507,70983,70657],{"class":513},[507,70985,70660],{"class":58306},[507,70987,70988],{"class":509,"line":61057},[507,70989,70990],{"class":562},"# @title Run Simulation & Display Results\n",[507,70992,70993],{"class":509,"line":61066},[507,70994,70995],{"class":58306},"import math\n",[507,70997,70998],{"class":509,"line":61076},[507,70999,71000],{"class":58306},"from typing import Tuple, Dict\n",[507,71002,71003],{"class":509,"line":61085},[507,71004,71005],{"class":58306},"import pandas as pd\n",[507,71007,71008],{"class":509,"line":61094},[507,71009,70824],{"class":58306},[507,71011,71012],{"class":509,"line":61100},[507,71013,71014],{"class":58306},"import matplotlib.pyplot as plt\n",[507,71016,71017],{"class":509,"line":61109},[507,71018,71019],{"class":58306},"from IPython.display import display, Markdown\n",[507,71021,71022],{"class":509,"line":61121},[507,71023,556],{"emptyLinePlaceholder":133},[507,71025,71026,71029,71031,71034,71036,71039,71042],{"class":509,"line":61134},[507,71027,71028],{"class":58306},"def wilson_ci",[507,71030,580],{"class":583},[507,71032,71033],{"class":58306},"k: int, n: int, z: float = 1.96",[507,71035,3649],{"class":583},[507,71037,71038],{"class":58306}," -> Tuple",[507,71040,71041],{"class":583},"[float, float]",[507,71043,1728],{"class":58306},[507,71045,71046,71049,71051,71054],{"class":509,"line":61146},[507,71047,71048],{"class":58306},"    if n == 0: return ",[507,71050,580],{"class":583},[507,71052,71053],{"class":58306},"0.0, 1.0",[507,71055,587],{"class":583},[507,71057,71058,71061,71063],{"class":509,"line":61151},[507,71059,71060],{"class":58306},"    p_hat, z2_n = k \u002F n, z",[507,71062,2377],{"class":583},[507,71064,71065],{"class":58306},"2 \u002F n\n",[507,71067,71068,71071,71073],{"class":509,"line":61156},[507,71069,71070],{"class":58306},"    denom = 1.0 ",[507,71072,2107],{"class":583},[507,71074,71075],{"class":58306}," z2_n\n",[507,71077,71078,71081,71083,71086,71088,71091,71093],{"class":509,"line":61161},[507,71079,71080],{"class":58306},"    center = ",[507,71082,580],{"class":583},[507,71084,71085],{"class":58306},"p_hat ",[507,71087,2107],{"class":583},[507,71089,71090],{"class":58306}," z2_n \u002F 2",[507,71092,3649],{"class":583},[507,71094,71095],{"class":58306}," \u002F denom\n",[507,71097,71098,71101,71103,71106,71108,71111,71113,71115,71117,71119,71122,71124,71127,71129,71132,71134,71137,71139,71142,71144,71147,71149,71151,71154],{"class":509,"line":61180},[507,71099,71100],{"class":58306},"    span = ",[507,71102,580],{"class":583},[507,71104,71105],{"class":58306},"z ",[507,71107,2391],{"class":583},[507,71109,71110],{"class":58306}," math.sqrt",[507,71112,580],{"class":583},[507,71114,71085],{"class":58306},[507,71116,2391],{"class":583},[507,71118,58644],{"class":583},[507,71120,71121],{"class":58306},"1 - p_hat",[507,71123,3649],{"class":583},[507,71125,71126],{"class":58306}," \u002F n ",[507,71128,2107],{"class":583},[507,71130,71131],{"class":58306}," z",[507,71133,2377],{"class":583},[507,71135,71136],{"class":58306},"2 \u002F ",[507,71138,580],{"class":583},[507,71140,71141],{"class":58306},"4 ",[507,71143,2391],{"class":583},[507,71145,71146],{"class":58306}," n",[507,71148,2377],{"class":583},[507,71150,584],{"class":58306},[507,71152,71153],{"class":583},")))",[507,71155,71095],{"class":58306},[507,71157,71158,71161,71163,71165,71167,71170,71172,71175,71177,71180,71182,71185],{"class":509,"line":61186},[507,71159,71160],{"class":58306},"    return ",[507,71162,580],{"class":583},[507,71164,36712],{"class":58306},[507,71166,580],{"class":583},[507,71168,71169],{"class":58306},"0.0, center - span",[507,71171,3649],{"class":583},[507,71173,71174],{"class":58306},", min",[507,71176,580],{"class":583},[507,71178,71179],{"class":58306},"1.0, center ",[507,71181,2107],{"class":583},[507,71183,71184],{"class":58306}," span",[507,71186,22540],{"class":583},[507,71188,71189],{"class":509,"line":61195},[507,71190,556],{"emptyLinePlaceholder":133},[507,71192,71193,71196,71198,71201,71203],{"class":509,"line":61202},[507,71194,71195],{"class":58306},"def shots_for_epsilon",[507,71197,580],{"class":583},[507,71199,71200],{"class":58306},"p: float, epsilon: float = 0.01, z: float = 1.96",[507,71202,3649],{"class":583},[507,71204,71205],{"class":58306}," -> int:\n",[507,71207,71208,71211,71213,71215,71217,71219],{"class":509,"line":61209},[507,71209,71210],{"class":58306},"    if p ",[507,71212,2391],{"class":583},[507,71214,58644],{"class":583},[507,71216,70770],{"class":58306},[507,71218,3649],{"class":583},[507,71220,71221],{"class":58306}," == 0: return 1\n",[507,71223,71224,71227,71229,71232,71234,71237,71239,71241,71243,71246,71248,71250,71252,71254,71256,71259,71262,71264,71266,71268,71270],{"class":509,"line":61217},[507,71225,71226],{"class":58306},"    return max",[507,71228,580],{"class":583},[507,71230,71231],{"class":58306},"1, int",[507,71233,580],{"class":583},[507,71235,71236],{"class":58306},"math.ceil",[507,71238,63480],{"class":583},[507,71240,666],{"class":58306},[507,71242,2377],{"class":583},[507,71244,71245],{"class":58306},"2 ",[507,71247,2391],{"class":583},[507,71249,68242],{"class":58306},[507,71251,2391],{"class":583},[507,71253,58644],{"class":583},[507,71255,70770],{"class":58306},[507,71257,71258],{"class":583},"))",[507,71260,71261],{"class":58306}," \u002F ",[507,71263,580],{"class":583},[507,71265,68163],{"class":58306},[507,71267,2377],{"class":583},[507,71269,584],{"class":58306},[507,71271,71272],{"class":583},"))))\n",[507,71274,71275],{"class":509,"line":61228},[507,71276,556],{"emptyLinePlaceholder":133},[507,71278,71279,71282,71284,71287,71289],{"class":509,"line":61239},[507,71280,71281],{"class":58306},"def summarize_model",[507,71283,580],{"class":583},[507,71285,71286],{"class":58306},"model, bits: np.ndarray, epsilons",[507,71288,3649],{"class":583},[507,71290,71291],{"class":58306}," -> Dict:\n",[507,71293,71294,71297,71299,71301],{"class":509,"line":61250},[507,71295,71296],{"class":58306},"    n = len",[507,71298,580],{"class":583},[507,71300,68846],{"class":58306},[507,71302,587],{"class":583},[507,71304,71305,71308,71310,71312],{"class":509,"line":61255},[507,71306,71307],{"class":58306},"    ones = sum",[507,71309,580],{"class":583},[507,71311,68846],{"class":58306},[507,71313,587],{"class":583},[507,71315,71316,71319,71321,71324],{"class":509,"line":61264},[507,71317,71318],{"class":58306},"    lo, hi = wilson_ci",[507,71320,580],{"class":583},[507,71322,71323],{"class":58306},"ones, n",[507,71325,587],{"class":583},[507,71327,71328],{"class":509,"line":61271},[507,71329,71330],{"class":58306},"    row = {\n",[507,71332,71333,71336,71338,71341,71343],{"class":509,"line":61278},[507,71334,71335],{"class":58306},"        \"Model\": model.name, \"p ",[507,71337,580],{"class":583},[507,71339,71340],{"class":58306},"target",[507,71342,3649],{"class":583},[507,71344,71345],{"class":58306},"\": model.satisfaction_prob, \"Shots\": n,\n",[507,71347,71348,71351,71353,71356,71358],{"class":509,"line":61286},[507,71349,71350],{"class":58306},"        \"Ones\": ones, \"p ",[507,71352,580],{"class":583},[507,71354,71355],{"class":58306},"observed",[507,71357,3649],{"class":583},[507,71359,71360],{"class":58306},"\": ones \u002F n if n > 0 else 0,\n",[507,71362,71363],{"class":509,"line":61307},[507,71364,71365],{"class":58306},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[507,71367,71368],{"class":509,"line":61318},[507,71369,59483],{"class":58306},[507,71371,71372],{"class":509,"line":61330},[507,71373,71374],{"class":58306},"    for eps in epsilons:\n",[507,71376,71377,71380,71383,71386,71388,71391],{"class":509,"line":61341},[507,71378,71379],{"class":58306},"        row",[507,71381,71382],{"class":583},"[f\"Shots for ±{eps:.1%}\"]",[507,71384,71385],{"class":58306}," = shots_for_epsilon",[507,71387,580],{"class":583},[507,71389,71390],{"class":58306},"model.satisfaction_prob, eps",[507,71392,587],{"class":583},[507,71394,71395],{"class":509,"line":61352},[507,71396,71397],{"class":58306},"    return row\n",[507,71399,71400],{"class":509,"line":61357},[507,71401,556],{"emptyLinePlaceholder":133},[507,71403,71404],{"class":509,"line":61362},[507,71405,71406],{"class":562},"# --- Run Simulation ---\n",[507,71408,71409,71412],{"class":509,"line":61367},[507,71410,71411],{"class":58306},"summary_rows = ",[507,71413,71414],{"class":583},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[507,71416,71417,71420,71422,71425],{"class":509,"line":61381},[507,71418,71419],{"class":58306},"summary_df = pd.DataFrame",[507,71421,580],{"class":583},[507,71423,71424],{"class":58306},"summary_rows",[507,71426,587],{"class":583},[507,71428,71429],{"class":509,"line":61393},[507,71430,71431],{"class":58306},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[507,71433,71434,71437,71440,71443,71445,71448,71451,71453],{"class":509,"line":61399},[507,71435,71436],{"class":58306},"    summary_df",[507,71438,71439],{"class":583},"[\"Cost (USD)\"]",[507,71441,71442],{"class":58306}," = FIXED_OVERHEAD_USD ",[507,71444,2107],{"class":583},[507,71446,71447],{"class":58306}," summary_df",[507,71449,71450],{"class":583},"[\"Shots\"]",[507,71452,8229],{"class":583},[507,71454,71455],{"class":58306}," COST_PER_SHOT_USD\n",[507,71457,71458],{"class":509,"line":61404},[507,71459,556],{"emptyLinePlaceholder":133},[507,71461,71462,71464,71466,71468,71470,71472],{"class":509,"line":61412},[507,71463,70479],{"class":58306},[507,71465,580],{"class":583},[507,71467,70484],{"class":58306},[507,71469,580],{"class":583},[507,71471,22281],{"class":58306},[507,71473,71474],{"class":562},"### Simulation Summary\"))\n",[507,71476,71477,71479,71481,71484,71486,71489,71491,71494,71496,71499],{"class":509,"line":61422},[507,71478,70479],{"class":58306},[507,71480,580],{"class":583},[507,71482,71483],{"class":58306},"summary_df.style.format",[507,71485,580],{"class":583},[507,71487,71488],{"class":58306},"precision=4",[507,71490,3649],{"class":583},[507,71492,71493],{"class":58306},".background_gradient",[507,71495,580],{"class":583},[507,71497,71498],{"class":58306},"cmap='viridis', subset=",[507,71500,71501],{"class":583},"['p (observed)', 'CI width']))\n",[507,71503,71504],{"class":509,"line":61427},[507,71505,556],{"emptyLinePlaceholder":133},[507,71507,71508],{"class":509,"line":61439},[507,71509,71510],{"class":562},"# --- Plotting ---\n",[507,71512,71513,71516,71518,71520],{"class":509,"line":61450},[507,71514,71515],{"class":58306},"plt.style.use",[507,71517,580],{"class":583},[507,71519,69716],{"class":58306},[507,71521,587],{"class":583},[507,71523,71524,71527,71529,71532,71534,71537,71539,71542,71544,71547],{"class":509,"line":61460},[507,71525,71526],{"class":58306},"fig, ",[507,71528,580],{"class":583},[507,71530,71531],{"class":58306},"ax1, ax2",[507,71533,3649],{"class":583},[507,71535,71536],{"class":58306}," = plt.subplots",[507,71538,580],{"class":583},[507,71540,71541],{"class":58306},"1, 2, figsize=",[507,71543,580],{"class":583},[507,71545,71546],{"class":58306},"16, 6",[507,71548,22540],{"class":583},[507,71550,71551],{"class":509,"line":61471},[507,71552,556],{"emptyLinePlaceholder":133},[507,71554,71555],{"class":509,"line":61476},[507,71556,71557],{"class":562},"# Plot 1\n",[507,71559,71560,71563,71566,71569],{"class":509,"line":61481},[507,71561,71562],{"class":58306},"y = summary_df",[507,71564,71565],{"class":583},"[\"p (observed)\"]",[507,71567,71568],{"class":58306},".to_numpy",[507,71570,781],{"class":583},[507,71572,71573,71576],{"class":509,"line":61493},[507,71574,71575],{"class":58306},"lower_error = y - summary_df",[507,71577,71578],{"class":583},"[\"CI 95% low\"]\n",[507,71580,71581,71584,71587],{"class":509,"line":61504},[507,71582,71583],{"class":58306},"upper_error = summary_df",[507,71585,71586],{"class":583},"[\"CI 95% high\"]",[507,71588,71589],{"class":58306}," - y\n",[507,71591,71592,71595],{"class":509,"line":61513},[507,71593,71594],{"class":58306},"yerr = np.vstack",[507,71596,71597],{"class":583},"([np.abs(lower_error), np.abs(upper_error)])\n",[507,71599,71600],{"class":509,"line":61524},[507,71601,556],{"emptyLinePlaceholder":133},[507,71603,71604,71607,71609,71612,71615,71618],{"class":509,"line":61529},[507,71605,71606],{"class":58306},"ax1.bar",[507,71608,580],{"class":583},[507,71610,71611],{"class":58306},"summary_df",[507,71613,71614],{"class":583},"[\"Model\"]",[507,71616,71617],{"class":58306},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[507,71619,587],{"class":583},[507,71621,71622,71625,71627,71630,71632,71635,71637,71640],{"class":509,"line":61534},[507,71623,71624],{"class":58306},"ax1.set_title",[507,71626,580],{"class":583},[507,71628,71629],{"class":58306},"\"Observed Success Rate ",[507,71631,580],{"class":583},[507,71633,71634],{"class":58306},"95% Wilson CI",[507,71636,3649],{"class":583},[507,71638,71639],{"class":58306},"\", fontsize=14",[507,71641,587],{"class":583},[507,71643,71644,71647,71649,71652],{"class":509,"line":61544},[507,71645,71646],{"class":58306},"ax1.tick_params",[507,71648,580],{"class":583},[507,71650,71651],{"class":58306},"axis='x', rotation=30, labelsize=10",[507,71653,587],{"class":583},[507,71655,71656],{"class":509,"line":61550},[507,71657,556],{"emptyLinePlaceholder":133},[507,71659,71660],{"class":509,"line":61564},[507,71661,71662],{"class":562},"# Plot 2\n",[507,71664,71665],{"class":509,"line":61569},[507,71666,71667],{"class":58306},"width = 0.2\n",[507,71669,71670,71673,71675,71677,71679,71682],{"class":509,"line":61579},[507,71671,71672],{"class":58306},"x = np.arange",[507,71674,580],{"class":583},[507,71676,1763],{"class":58306},[507,71678,580],{"class":583},[507,71680,71681],{"class":58306},"MODELS",[507,71683,22540],{"class":583},[507,71685,71686,71689],{"class":509,"line":61584},[507,71687,71688],{"class":58306},"colors = ",[507,71690,71691],{"class":583},"['#ff9999','#66b3ff','#99ff99']\n",[507,71693,71694,71697,71699,71702,71704],{"class":509,"line":61596},[507,71695,71696],{"class":58306},"for i, eps in enumerate",[507,71698,580],{"class":583},[507,71700,71701],{"class":58306},"EPSILONS",[507,71703,3649],{"class":583},[507,71705,1728],{"class":58306},[507,71707,71708,71711],{"class":509,"line":61601},[507,71709,71710],{"class":58306},"    req = summary_df",[507,71712,71713],{"class":583},"[f\"Shots for ±{eps:.1%}\"]\n",[507,71715,71716,71719,71721,71723,71725,71727,71729,71732,71735,71738],{"class":509,"line":61606},[507,71717,71718],{"class":58306},"    ax2.bar",[507,71720,580],{"class":583},[507,71722,69745],{"class":58306},[507,71724,2107],{"class":583},[507,71726,8246],{"class":58306},[507,71728,2391],{"class":583},[507,71730,71731],{"class":58306}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[507,71733,71734],{"class":583},"[i]",[507,71736,71737],{"class":58306},", edgecolor='black'",[507,71739,587],{"class":583},[507,71741,71742],{"class":509,"line":61629},[507,71743,556],{"emptyLinePlaceholder":133},[507,71745,71746,71749,71751,71753,71755,71758],{"class":509,"line":61634},[507,71747,71748],{"class":58306},"ax2.set_xticks",[507,71750,580],{"class":583},[507,71752,69745],{"class":58306},[507,71754,2107],{"class":583},[507,71756,71757],{"class":58306}," width, summary_df",[507,71759,71760],{"class":583},"[\"Model\"])\n",[507,71762,71763,71766,71768,71771,71773,71776,71778,71780],{"class":509,"line":61657},[507,71764,71765],{"class":58306},"ax2.set_ylabel",[507,71767,580],{"class":583},[507,71769,71770],{"class":58306},"\"Required Shots ",[507,71772,580],{"class":583},[507,71774,71775],{"class":58306},"Log Scale",[507,71777,3649],{"class":583},[507,71779,22281],{"class":58306},[507,71781,587],{"class":583},[507,71783,71784,71787,71789,71791],{"class":509,"line":61668},[507,71785,71786],{"class":58306},"ax2.set_yscale",[507,71788,580],{"class":583},[507,71790,70376],{"class":58306},[507,71792,587],{"class":583},[507,71794,71795,71798,71800,71803],{"class":509,"line":61707},[507,71796,71797],{"class":58306},"ax2.set_title",[507,71799,580],{"class":583},[507,71801,71802],{"class":58306},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[507,71804,587],{"class":583},[507,71806,71807,71810,71812,71814],{"class":509,"line":61717},[507,71808,71809],{"class":58306},"ax2.tick_params",[507,71811,580],{"class":583},[507,71813,71651],{"class":58306},[507,71815,587],{"class":583},[507,71817,71818,71821,71823,71826],{"class":509,"line":61729},[507,71819,71820],{"class":58306},"ax2.legend",[507,71822,580],{"class":583},[507,71824,71825],{"class":58306},"title=\"Tolerance\"",[507,71827,587],{"class":583},[507,71829,71830],{"class":509,"line":61749},[507,71831,556],{"emptyLinePlaceholder":133},[507,71833,71834,71837],{"class":509,"line":61767},[507,71835,71836],{"class":58306},"plt.tight_layout",[507,71838,781],{"class":583},[507,71840,71841,71844],{"class":509,"line":61772},[507,71842,71843],{"class":58306},"plt.show",[507,71845,781],{"class":583},[507,71847,71848],{"class":509,"line":61777},[507,71849,70660],{"class":58306},[507,71851,71852],{"class":509,"line":61789},[507,71853,556],{"emptyLinePlaceholder":133},[507,71855,71856,71859,71861,71864,71867],{"class":509,"line":61799},[507,71857,71858],{"class":517},"nb ",[507,71860,573],{"class":572},[507,71862,71863],{"class":517}," nbf.v4.",[507,71865,71866],{"class":576},"new_notebook",[507,71868,781],{"class":517},[507,71870,71871,71874,71877,71879,71881],{"class":509,"line":61809},[507,71872,71873],{"class":517},"nb[",[507,71875,71876],{"class":730},"\"cells\"",[507,71878,8206],{"class":517},[507,71880,573],{"class":572},[507,71882,2177],{"class":517},[507,71884,71885,71888,71891],{"class":509,"line":61814},[507,71886,71887],{"class":517},"    nbf.v4.",[507,71889,71890],{"class":576},"new_markdown_cell",[507,71892,71893],{"class":517},"(md_intro),\n",[507,71895,71896,71898,71901],{"class":509,"line":61819},[507,71897,71887],{"class":517},[507,71899,71900],{"class":576},"new_code_cell",[507,71902,71903],{"class":517},"(code_config),\n",[507,71905,71906,71908,71910],{"class":509,"line":61827},[507,71907,71887],{"class":517},[507,71909,71900],{"class":576},[507,71911,71912],{"class":517},"(code_lib_and_run),\n",[507,71914,71915],{"class":509,"line":61837},[507,71916,1794],{"class":517},[507,71918,71919],{"class":509,"line":61852},[507,71920,556],{"emptyLinePlaceholder":133},[507,71922,71923,71926,71928],{"class":509,"line":61863},[507,71924,71925],{"class":517},"nb_path ",[507,71927,573],{"class":572},[507,71929,71930],{"class":730}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[507,71932,71933,71935,71937,71940,71942,71944,71947,71949,71952,71954,71956],{"class":509,"line":61875},[507,71934,69626],{"class":513},[507,71936,69629],{"class":572},[507,71938,71939],{"class":517},"(nb_path, ",[507,71941,69639],{"class":730},[507,71943,622],{"class":517},[507,71945,71946],{"class":2155},"encoding",[507,71948,573],{"class":572},[507,71950,71951],{"class":730},"\"utf-8\"",[507,71953,655],{"class":517},[507,71955,521],{"class":513},[507,71957,69646],{"class":517},[507,71959,71960,71963,71966],{"class":509,"line":61880},[507,71961,71962],{"class":517},"    nbf.",[507,71964,71965],{"class":576},"write",[507,71967,71968],{"class":517},"(nb, f)\n",[507,71970,71971],{"class":509,"line":61887},[507,71972,556],{"emptyLinePlaceholder":133},[507,71974,71975],{"class":509,"line":61896},[507,71976,71977],{"class":562},"# ---------- Final confirmation ----------\n",[507,71979,71980,71982,71984,71986,71988,71990,71992,71995,71997,71999],{"class":509,"line":61910},[507,71981,8525],{"class":572},[507,71983,580],{"class":517},[507,71985,22281],{"class":730},[507,71987,61723],{"class":572},[507,71989,22281],{"class":730},[507,71991,8313],{"class":572},[507,71993,71994],{"class":730}," \"=\"",[507,71996,2391],{"class":572},[507,71998,48980],{"class":583},[507,72000,587],{"class":517},[507,72002,72003,72005,72007,72010],{"class":509,"line":61921},[507,72004,8525],{"class":572},[507,72006,580],{"class":517},[507,72008,72009],{"class":730},"\"SCRIPT COMPLETE\"",[507,72011,587],{"class":517},[507,72013,72014,72016,72018,72021,72023,72025],{"class":509,"line":61933},[507,72015,8525],{"class":572},[507,72017,580],{"class":517},[507,72019,72020],{"class":730},"\"=\"",[507,72022,2391],{"class":572},[507,72024,48980],{"class":583},[507,72026,587],{"class":517},[507,72028,72029,72031,72033,72035,72038,72040,72043,72045,72047],{"class":509,"line":61938},[507,72030,8525],{"class":572},[507,72032,580],{"class":517},[507,72034,22278],{"class":513},[507,72036,72037],{"class":730},"\"Google Colab notebook saved to: ",[507,72039,2810],{"class":583},[507,72041,72042],{"class":517},"nb_path",[507,72044,2872],{"class":583},[507,72046,22281],{"class":730},[507,72048,587],{"class":517},[507,72050,72051,72053,72055,72057,72060,72062,72065,72067,72069],{"class":509,"line":61943},[507,72052,8525],{"class":572},[507,72054,580],{"class":517},[507,72056,22278],{"class":513},[507,72058,72059],{"class":730},"\"Summary data saved to: ",[507,72061,2810],{"class":583},[507,72063,72064],{"class":517},"summary_csv_path",[507,72066,2872],{"class":583},[507,72068,22281],{"class":730},[507,72070,587],{"class":517},[507,72072,72073,72075,72077,72079,72082,72084,72087,72089,72091,72093,72096,72098,72100],{"class":509,"line":61953},[507,72074,8525],{"class":572},[507,72076,580],{"class":517},[507,72078,22278],{"class":513},[507,72080,72081],{"class":730},"\"Plots saved to: ",[507,72083,2810],{"class":583},[507,72085,72086],{"class":517},"fig1_path",[507,72088,2872],{"class":583},[507,72090,622],{"class":730},[507,72092,2810],{"class":583},[507,72094,72095],{"class":517},"fig2_path",[507,72097,2872],{"class":583},[507,72099,22281],{"class":730},[507,72101,587],{"class":517},[498,72103,72106],{"className":72104,"code":72105,"language":7039,"meta":104},[8531],"Running simulation...\nSaved data to CSV and JSON files.\n",[504,72107,72105],{"__ignoreMap":104},[831,72109],{"alt":58383,"src":72110},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[831,72112],{"alt":63289,"src":72113},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[498,72115,72118],{"className":72116,"code":72117,"language":7039,"meta":104},[8531],"Generated and saved plots.\n",[504,72119,72117],{"__ignoreMap":104},[498,72121,72124],{"className":72122,"code":72123,"language":7039,"meta":104},[8531],"==================================================\nSCRIPT COMPLETE\n==================================================\nGoogle Colab notebook saved to: Sampling_Budget_Qubit_Bitstrings.ipynb\nSummary data saved to: summary.csv\nPlots saved to: fig_ones_fraction.png, fig_required_shots.png\n",[504,72125,72123],{"__ignoreMap":104},[72127,72128],"hr",{},[13,72130,41556],{"id":41555},[27370,72132,72133,72140,72147,72154,72161,72168,72175,72182,72189,72196,72203,72210,72217,72228,72235,72242],{},[45,72134,72135,72136,72139],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[1031,72137,72138],{},"The American Statistician",", 52(2), pp. 119-126.",[45,72141,72142,72143,72146],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[1031,72144,72145],{},"Statistical Science",", 16(2), pp. 101-133.",[45,72148,72149,72150,72153],{},"Cochran, W.G. (1977) ",[1031,72151,72152],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[45,72155,72156,72157,72160],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[1031,72158,72159],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[45,72162,72163,72164,72167],{},"Harris, C.R., Millman, K.J., van der Walt, S.J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M.H., Brett, M., Haldane, A., del Río, J.F., Wiebe, M., Peterson, P., Gérard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C. and Oliphant, T.E. (2020) 'Array programming with NumPy', ",[1031,72165,72166],{},"Nature",", 585, pp. 357–362.",[45,72169,72170,72171,72174],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[1031,72172,72173],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[45,72176,72177,72178,72181],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[1031,72179,72180],{},"Annals of Physics",", 303(1), pp. 2-30.",[45,72183,72184,72185,72188],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[1031,72186,72187],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[45,72190,72191,72192,72195],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[1031,72193,72194],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[45,72197,72198,72199,72202],{},"Newman, M. (2013) ",[1031,72200,72201],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[45,72204,72205,72206,72209],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[1031,72207,72208],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[45,72211,72212,72213,72216],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[1031,72214,72215],{},"Quantum",", 2, p. 79.",[45,72218,72219,72220,72223,72224,53],{},"The pandas development team (2020) ",[1031,72221,72222],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[49,72225,72227],{"href":72226},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[45,72229,72230,72231,72234],{},"Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, İ., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P. and SciPy 1.0 Contributors (2020) 'SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python', ",[1031,72232,72233],{},"Nature Methods",", 17, pp. 261–272.",[45,72236,72237,72238,72241],{},"Wilde, M.M. (2017) ",[1031,72239,72240],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[45,72243,72244,72245,72248],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[1031,72246,72247],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[13,72250,940],{"id":939},[18,72252,41598,72253,947,72255,53],{},[49,72254,41670],{"href":67686},[49,72256,67634],{"href":72257},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[953,72259,72260],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":72262},[72263,72264],{"id":41555,"depth":105,"text":41556},{"id":939,"depth":105,"text":940},[112,969,970,72266],"Sampling budget",[72268],{"username":467,"name":974,"role":975,"bio":976,"links":72269},[72270,72271],{"label":979,"href":980},{"label":982,"href":67686},{"username":467,"name":974,"role":984},"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. A notebook by Onri Jay Benally, republished with permission.",{},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications","4 min read",[],{"title":67676,"description":72273},"blog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications",[67672,143],"HETemA9RSbsiqyryZFFLOYbrY0iNodc-iLIt3qtwByA",{"id":72284,"title":72285,"authors":72286,"body":72287,"breadcrumb":72410,"builders":72412,"byline":116,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":72413,"description":72414,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":116,"lessonCount":116,"meta":72415,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":72416,"publishDate":72417,"readingTime":116,"related":72418,"relatedProjects":116,"seo":72419,"stem":72422,"tags":72423,"track":116,"trackName":116,"__hash__":72425},"blog\u002Fblog\u002Fchangelog.md","What's new on Qollab",[8],{"type":10,"value":72288,"toc":72408},[72289,72298,72314,72326,72342,72353,72360,72367,72375,72382,72390,72398],[72290,72291,72295],"log-entry",{"date":72292,"tag":72293,"title":72294},"July 10, 2026","Backends","AWS Braket local simulator",[18,72296,72297],{},"The AWS Braket local simulator is selectable as a backend, alongside qiskit-aer.",[72290,72299,72303],{"date":72300,"tag":72301,"title":72302},"July 5, 2026","Performance","Playground simulation ~2.6x faster",[18,72304,72305,72306,72309,72310,72313],{},"qiskit-aer was being compiled with ",[504,72307,72308],{},"-Oz",". Rebuilding it with ",[504,72311,72312],{},"-O2"," made statevector simulation about 2.67x faster, and Aer's AVX2 kernels now run on wasm through Emscripten's SIMD128 emulation.",[72290,72315,72319],{"date":72316,"tag":72317,"title":72318},"June 30, 2026","Runtime","Python 3.14 in the browser",[18,72320,72321,72322,72325],{},"The in-browser runtime moved from Pyodide 0.29.4 to 314.0.1: Python 3.14, Emscripten 5.0.3, and the ",[504,72323,72324],{},"pyemscripten_2026"," wheel set.",[72290,72327,72331,72334],{"date":72328,"tag":72329,"title":72330},"June 29, 2026","Community","Forum and project discussions",[18,72332,72333],{},"The Qollab forum is live, and every project page has a Community tab backed by it.",[42,72335,72336,72339],{},[45,72337,72338],{},"Discourse-backed, with avatars synced from your Qollab profile.",[45,72340,72341],{},"Author-deleted posts are hidden from search and excluded from comment counts.",[72290,72343,72347],{"date":72344,"tag":72345,"title":72346},"June 22, 2026","Profiles","Contributor directory and platform stats",[18,72348,72349,72352],{},[504,72350,72351],{},"\u002Fcontributor"," lists everyone building on Qollab. The homepage carries platform stats and a top-creator leaderboard, and both the gallery and profile pages paginate.",[72290,72354,72357],{"date":72355,"tag":72317,"title":72356},"June 9, 2026","qiskit-aer runs in the browser",[18,72358,72359],{},"qiskit-aer is built as a Pyodide\u002FEmscripten wheel and linked against OpenBLAS, so simulation runs locally with no backend round-trip. High-depth circuits now warn before you run them.",[72290,72361,72364],{"date":72362,"tag":72293,"title":72363},"June 6, 2026","IBM simulator backends",[18,72365,72366],{},"IBM simulators are selectable as backends, alongside the Qiskit sampler.",[72290,72368,72372],{"date":72369,"tag":72370,"title":72371},"May 13, 2026","Editor","Autosave and coauthors",[18,72373,72374],{},"Project content and code autosave as you work. Projects can list coauthors, so multi-builder work credits everyone.",[72290,72376,72379],{"date":72377,"tag":72370,"title":72378},"April 26, 2026","Video embeds in project write-ups",[18,72380,72381],{},"Project write-ups support YouTube and self-hosted video embeds.",[72290,72383,72387],{"date":72384,"tag":72385,"title":72386},"April 22, 2026","Playground","Run without an account",[18,72388,72389],{},"Circuits run without signing in. Sign in when you want to save or publish.",[72290,72391,72395],{"date":72392,"tag":72393,"title":72394},"February 22, 2026","Hardware","Runs on real quantum hardware",[18,72396,72397],{},"Circuits run on real quantum hardware, not only simulators.",[94,72399,72405],{"dark":104,"f1":72400,"f2":72401,"l1":72402,"l2":72403,"title":72404},"https:\u002F\u002Fqollab.xyz","https:\u002F\u002Fgithub.com\u002Fqollabxyz","Open the Playground","Follow on GitHub","Releases ship every week or two.",[18,72406,72407],{},"Source, issues, and release tags are on GitHub.",{"title":104,"searchDepth":105,"depth":105,"links":72409},[],[112,969,72411],"Product updates",[],"What shipped, when. Runtime and backend changes, Playground performance, and project tooling.","What shipped on Qollab: runtime and backend changes, Playground performance, the community forum, and project tooling. Updated as we release.",{},"\u002Fblog\u002Fchangelog","2026-07-10",[],{"title":72420,"description":72421},"Qollab changelog: runtime, backends, and Playground updates","What shipped on Qollab: real-hardware runs, in-browser qiskit-aer, Pyodide 314 \u002F Python 3.14, IBM and AWS Braket backends, and the community forum.","blog\u002Fchangelog",[72424,24427],"changelog","D_cHhQDBpGSqWXtUTTiXrjqC1-ovSNLZ7AaKHHqzdvY",{"id":72427,"title":72428,"authors":72429,"body":72431,"breadcrumb":73051,"builders":73053,"byline":73074,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":73078,"description":73079,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":73081,"hero":73083,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":73086,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":73087,"publishDate":73088,"readingTime":993,"related":73089,"relatedProjects":73090,"seo":73106,"stem":73109,"tags":73110,"track":116,"trackName":116,"__hash__":73113},"blog\u002Fblog\u002Fentangled-body.md","Project Showcase: Entangled Body",[72430],"lukeshim",{"type":10,"value":72432,"toc":73044},[72433,72436,72439,72442,72450,72454,72457,72464,72467,72471,72474,72479,72487,72492,72496,72499,72504,72507,72991,72996,72999,73003,73006,73011,73019,73024,73028,73031,73041],[18,72434,72435],{},"Entangled Body is a point-cloud human figure whose parts respond to each other across distance. Touch one region and somewhere else reacts; move your viewpoint and the body reveals itself differently.",[18,72437,72438],{},"It is a quantum-inspired artwork, not a science diagram. Body regions are mapped to qubits, and the outputs of quantum circuits, measured bits, count distributions, correlations, drive how the body activates, how strongly regions connect, and how it resolves from scattered particles into a more stable form.",[18,72440,72441],{},"It is a project from Qollab's Creative Challenge, built by Chanhyuk Park and Luke Shim, who came to quantum from the art and computer-science sides rather than physics.",[72443,72444,72447],"pull-quote",{"avatar":104,"name":72445,"role":72446,"username":72430},"Luke Shim","Developer, Entangled Body",[18,72448,72449],{},"Once I built my own BB84 simulator and could see how measurement, basis choice, and eavesdropping affected outcomes, quantum mechanics stopped feeling like abstract physics and started feeling like a new computational medium.",[13,72451,72453],{"id":72452},"an-unseen-connection","An unseen connection",[18,72455,72456],{},"For Chanhyuk, the way in wasn't an equation. It was a memory of a traditional Chinese medicine clinic, and the strange logic of treating one part of the body by touching another.",[72443,72458,72461],{"avatar":104,"name":72459,"role":72460,"username":104},"Chanhyuk Park","Project lead, Entangled Body",[18,72462,72463],{},"I went because my wrist hurt, but the practitioner placed an acupuncture needle around my ankle. I am not making a medical claim from that experience, but it stayed with me as a metaphor for unseen connections inside the body.",[18,72465,72466],{},"Entanglement gave that intuition a name: a way to think about invisible relationships between separate things. The project grew out of that pairing, a felt idea about the body, and a quantum phenomenon that made it concrete.",[13,72468,72470],{"id":72469},"a-body-made-of-relations","A body made of relations",[18,72472,72473],{},"The figure they chose is a point-cloud astronaut on the moon, rendered as drifting particles rather than a solid form.",[72443,72475,72476],{"avatar":104,"name":72459,"role":72460,"username":104},[18,72477,72478],{},"An astronaut cannot exist alone in space; the body depends on the suit, life-support systems, communication signals, and the surrounding environment. Rendered as a point cloud, the astronaut becomes a temporary body made of data and particles, present but constantly dissolving and re-forming.",[18,72480,72481,72482,72486],{},"The visual language borrows from the artist ",[49,72483,72485],{"href":72484},"https:\u002F\u002Farchive.bridgesmathart.org\u002F2010\u002Fbridges2010-3.pdf","Julian Voss-Andreae",", whose sculptures appear or disappear depending on where you stand. That idea, that observation changes what becomes visible, is the heart of the piece.",[72443,72488,72489],{"avatar":104,"name":72459,"role":72460,"username":104},[18,72490,72491],{},"Entangled Body also explores how observation changes what becomes visible. We did not want it to feel like a science diagram; we wanted it to feel like an artwork first.",[13,72493,72495],{"id":72494},"how-the-quantum-drives-the-body","How the quantum drives the body",[18,72497,72498],{},"Under the hood, body regions map to qubits, and the circuit's outputs, measured bits, count distributions, and correlations, drive region activation, connection strength, and the way the body resolves from scattered particles into a stabler form. The quantum layer isn't only a theme; it shapes what you see. And they ran it on real hardware rather than a clean simulator.",[831,72500],{"caption":72501,"no":835,"poster":72502,"video":72503},"A touch on the point cloud rippling through the body graph as the circuit re-runs on real hardware. Press play, sound on.","\u002F_content\u002Fimages\u002Fentangled-body\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F960a7fe3-a278-4e61-8478-cc142d11f9a9",[18,72505,72506],{},"The function that turns a touch into a circuit is compact enough to read in one sitting:",[493,72508,72512],{"name":72509,"run-href":72510,"tag":72511},"entangled_body_demo.py","\u002Fu\u002Flukeshim\u002Fentangled-body","Python · excerpt",[498,72513,72516],{"className":500,"code":72514,"language":502,"meta":72515,"style":104},"# How a touch becomes a circuit. Excerpt from entangled_body_demo.py.\nfrom qiskit import QuantumCircuit\n\ndef build_ops(region, intensity, interaction):\n    \"\"\"Ops for the graph-collapse circuit anchored on the touched region.\"\"\"\n    observed = region if region in QUBIT_OF else \"torso\"\n    distances = spatial_graph_distances(observed)   # Dijkstra through the body graph\n    max_distance = max(d for d in distances.values() if d != float(\"inf\"))\n    ops = []\n    for rid, qubit, _ in REGIONS:\n        prob = _target_probability(observed, rid, distances[rid], max_distance, intensity, interaction)\n        ops.append((\"ry\", qubit, _prob_to_ry(prob)))\n    for src, tgt, strength in _ranked_links(distances, interaction):\n        s = max(0.05, min(1.0, strength))\n        ops.append((\"rzz\", QUBIT_OF[src], QUBIT_OF[tgt], _edge_angle(s, interaction)))\n    return ops\n\ndef build_circuit(ops, measure=True):\n    qc = QuantumCircuit(QUBIT_COUNT, QUBIT_COUNT)\n    for op in ops:\n        if op[0] == \"ry\":\n            qc.ry(op[2], op[1])\n        elif op[0] == \"rzz\":\n            qc.rzz(op[3], op[1], op[2])\n    if measure:\n        qc.measure(range(QUBIT_COUNT), range(QUBIT_COUNT))\n    return qc\n","· excerpt",[504,72517,72518,72523,72533,72537,72561,72566,72590,72606,72647,72656,72670,72683,72717,72732,72757,72795,72802,72806,72829,72857,72869,72887,72906,72924,72944,72951,72984],{"__ignoreMap":104},[507,72519,72520],{"class":509,"line":510},[507,72521,72522],{"class":562},"# How a touch becomes a circuit. Excerpt from entangled_body_demo.py.\n",[507,72524,72525,72527,72529,72531],{"class":509,"line":105},[507,72526,529],{"class":513},[507,72528,532],{"class":517},[507,72530,514],{"class":513},[507,72532,537],{"class":517},[507,72534,72535],{"class":509,"line":540},[507,72536,556],{"emptyLinePlaceholder":133},[507,72538,72539,72541,72544,72546,72549,72551,72554,72556,72559],{"class":509,"line":553},[507,72540,1370],{"class":513},[507,72542,72543],{"class":576}," build_ops",[507,72545,580],{"class":517},[507,72547,72548],{"class":1382},"region",[507,72550,622],{"class":517},[507,72552,72553],{"class":1382},"intensity",[507,72555,622],{"class":517},[507,72557,72558],{"class":1382},"interaction",[507,72560,1883],{"class":517},[507,72562,72563],{"class":509,"line":559},[507,72564,72565],{"class":730},"    \"\"\"Ops for the graph-collapse circuit anchored on the touched 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",[507,72612,573],{"class":572},[507,72614,65241],{"class":572},[507,72616,72617],{"class":517},"(d ",[507,72619,1630],{"class":513},[507,72621,25262],{"class":517},[507,72623,1636],{"class":513},[507,72625,72626],{"class":517}," distances.",[507,72628,72629],{"class":576},"values",[507,72631,1677],{"class":517},[507,72633,1645],{"class":513},[507,72635,25262],{"class":517},[507,72637,22619],{"class":572},[507,72639,59620],{"class":572},[507,72641,580],{"class":517},[507,72643,72644],{"class":730},"\"inf\"",[507,72646,22540],{"class":517},[507,72648,72649,72652,72654],{"class":509,"line":634},[507,72650,72651],{"class":517},"    ops ",[507,72653,573],{"class":572},[507,72655,1910],{"class":517},[507,72657,72658,72660,72663,72665,72668],{"class":509,"line":661},[507,72659,1916],{"class":513},[507,72661,72662],{"class":517}," rid, qubit, _ ",[507,72664,1636],{"class":513},[507,72666,72667],{"class":583}," REGIONS",[507,72669,1728],{"class":517},[507,72671,72672,72675,72677,72680],{"class":509,"line":678},[507,72673,72674],{"class":517},"        prob ",[507,72676,573],{"class":572},[507,72678,72679],{"class":576}," _target_probability",[507,72681,72682],{"class":517},"(observed, rid, distances[rid], max_distance, intensity, interaction)\n",[507,72684,72685,72688,72690,72692,72695,72698,72701,72704],{"class":509,"line":683},[507,72686,72687],{"class":517},"        ops.",[507,72689,1939],{"class":576},[507,72691,63480],{"class":517},[507,72693,72694],{"class":730},"\"ry\"",[507,72696,72697],{"class":517},", qubit, ",[507,72699,72700],{"class":576},"_prob_to_ry",[507,72702,72703],{"class":517},"(prob)))",[507,72705,72708,72709],{"class":72706,"tabindex":72707},"qg-help",0,"?",[507,72710,72713,72716],{"class":72711,"role":72712},"qg-help__tip","tooltip",[154,72714,72715],{},"One qubit per region."," The Ry angle sets how \"on\" that region should be: near-certain at the touched point, less certain the further it sits in the anatomical graph.",[507,72718,72719,72721,72724,72726,72729],{"class":509,"line":697},[507,72720,1916],{"class":513},[507,72722,72723],{"class":517}," src, tgt, strength ",[507,72725,1636],{"class":513},[507,72727,72728],{"class":576}," _ranked_links",[507,72730,72731],{"class":517},"(distances, interaction):\n",[507,72733,72734,72737,72739,72741,72743,72746,72748,72750,72752,72754],{"class":509,"line":710},[507,72735,72736],{"class":517},"        s ",[507,72738,573],{"class":572},[507,72740,65241],{"class":572},[507,72742,580],{"class":517},[507,72744,72745],{"class":583},"0.05",[507,72747,622],{"class":517},[507,72749,25230],{"class":572},[507,72751,580],{"class":517},[507,72753,57927],{"class":583},[507,72755,72756],{"class":517},", strength))\n",[507,72758,72759,72761,72763,72765,72768,72770,72773,72776,72778,72781,72784,72787],{"class":509,"line":715},[507,72760,72687],{"class":517},[507,72762,1939],{"class":576},[507,72764,63480],{"class":517},[507,72766,72767],{"class":730},"\"rzz\"",[507,72769,622],{"class":517},[507,72771,72772],{"class":583},"QUBIT_OF",[507,72774,72775],{"class":517},"[src], ",[507,72777,72772],{"class":583},[507,72779,72780],{"class":517},"[tgt], ",[507,72782,72783],{"class":576},"_edge_angle",[507,72785,72786],{"class":517},"(s, interaction)))",[507,72788,72708,72789],{"class":72706,"tabindex":72707},[507,72790,72791,72794],{"class":72711,"role":72712},[154,72792,72793],{},"Anatomical entanglement."," Rzz couples two qubits with a strength taken from the real body graph: head-to-chest is a strong link, torso-to-left-foot has no direct edge at all.",[507,72796,72797,72799],{"class":509,"line":721},[507,72798,2504],{"class":513},[507,72800,72801],{"class":517}," ops\n",[507,72803,72804],{"class":509,"line":736},[507,72805,556],{"emptyLinePlaceholder":133},[507,72807,72808,72810,72813,72815,72818,72820,72823,72825,72827],{"class":509,"line":748},[507,72809,1370],{"class":513},[507,72811,72812],{"class":576}," build_circuit",[507,72814,580],{"class":517},[507,72816,72817],{"class":1382},"ops",[507,72819,622],{"class":517},[507,72821,72822],{"class":1382},"measure",[507,72824,573],{"class":517},[507,72826,13878],{"class":583},[507,72828,1883],{"class":517},[507,72830,72831,72834,72836,72838,72840,72843,72845,72847,72849],{"class":509,"line":761},[507,72832,72833],{"class":517},"    qc ",[507,72835,573],{"class":572},[507,72837,577],{"class":576},[507,72839,580],{"class":517},[507,72841,72842],{"class":583},"QUBIT_COUNT",[507,72844,622],{"class":517},[507,72846,72842],{"class":583},[507,72848,3649],{"class":517},[507,72850,72708,72851],{"class":72706,"tabindex":72707},[507,72852,72853,72856],{"class":72711,"role":72712},[154,72854,72855],{},"14 qubits, 14 regions."," Head to left foot, every body part the installation tracks lives in one entangled circuit, not 14 separate ones.",[507,72858,72859,72861,72864,72866],{"class":509,"line":775},[507,72860,1916],{"class":513},[507,72862,72863],{"class":517}," op ",[507,72865,1636],{"class":513},[507,72867,72868],{"class":517}," ops:\n",[507,72870,72871,72873,72876,72878,72880,72882,72885],{"class":509,"line":784},[507,72872,1734],{"class":513},[507,72874,72875],{"class":517}," op[",[507,72877,601],{"class":583},[507,72879,8206],{"class":517},[507,72881,1723],{"class":572},[507,72883,72884],{"class":730}," \"ry\"",[507,72886,1728],{"class":517},[507,72888,72889,72892,72894,72897,72899,72902,72904],{"class":509,"line":796},[507,72890,72891],{"class":517},"            qc.",[507,72893,639],{"class":576},[507,72895,72896],{"class":517},"(op[",[507,72898,584],{"class":583},[507,72900,72901],{"class":517},"], op[",[507,72903,625],{"class":583},[507,72905,68725],{"class":517},[507,72907,72908,72911,72913,72915,72917,72919,72922],{"class":509,"line":809},[507,72909,72910],{"class":513},"        elif",[507,72912,72875],{"class":517},[507,72914,601],{"class":583},[507,72916,8206],{"class":517},[507,72918,1723],{"class":572},[507,72920,72921],{"class":730}," \"rzz\"",[507,72923,1728],{"class":517},[507,72925,72926,72928,72930,72932,72934,72936,72938,72940,72942],{"class":509,"line":1352},[507,72927,72891],{"class":517},[507,72929,20984],{"class":576},[507,72931,72896],{"class":517},[507,72933,8226],{"class":583},[507,72935,72901],{"class":517},[507,72937,625],{"class":583},[507,72939,72901],{"class":517},[507,72941,584],{"class":583},[507,72943,68725],{"class":517},[507,72945,72946,72948],{"class":509,"line":1357},[507,72947,1717],{"class":513},[507,72949,72950],{"class":517}," measure:\n",[507,72952,72953,72956,72958,72960,72962,72964,72966,72968,72970,72972,72974,72976],{"class":509,"line":1362},[507,72954,72955],{"class":517},"        qc.",[507,72957,72822],{"class":576},[507,72959,580],{"class":517},[507,72961,2204],{"class":572},[507,72963,580],{"class":517},[507,72965,72842],{"class":583},[507,72967,2213],{"class":517},[507,72969,2204],{"class":572},[507,72971,580],{"class":517},[507,72973,72842],{"class":583},[507,72975,71258],{"class":517},[507,72977,72708,72978],{"class":72706,"tabindex":72707},[507,72979,72980,72983],{"class":72711,"role":72712},[154,72981,72982],{},"The noise stays in."," Run on real IonQ hardware, this measurement carries genuine hardware noise. Luke and Chanhyuk kept it rather than smoothing it away, since the body is meant to always be becoming, not fixed.",[507,72985,72986,72988],{"class":509,"line":1367},[507,72987,2504],{"class":513},[507,72989,72990],{"class":517}," qc\n",[72443,72992,72993],{"avatar":104,"name":72445,"role":72446,"username":72430},[18,72994,72995],{},"For Entangled Body, that unpredictability was actually valuable. The project explores the idea that a digital body is constantly becoming rather than remaining fixed. Hardware noise and probabilistic measurement outcomes contributed to that feeling. Instead of treating quantum uncertainty as a problem, we treated it as part of the artistic and interactive experience.",[18,72997,72998],{},"That is the move that makes the piece more than a fancy random-number generator: the noise isn't hidden or normalized away. The body is always becoming, dissolving and re-forming, precisely because the measurements vary from run to run.",[13,73000,73002],{"id":73001},"where-it-could-go","Where it could go",[18,73004,73005],{},"Entangled Body is open for forking, and both builders want to push the connection between quantum behavior and the body further. For Luke, the direction is structural.",[72443,73007,73008],{"avatar":104,"name":72445,"role":72446,"username":72430},[18,73009,73010],{},"The core idea is treating quantum mechanics as part of the engine itself rather than simply a theme. I'd love to see multi-user entangled bodies that influence each other through shared quantum states, or artwork that continuously responds to live quantum measurements.",[18,73012,73013,73014,73018],{},"Chanhyuk wants it to leave the screen entirely, with gesture input, motion tracking, or spatial sound, the audience's own body could begin to affect the point-cloud body and become part of the work, turning it into an interactive installation or performance. For developers coming from outside quantum, his advice is to start visual and small: tools like ",[49,73015,73017],{"href":73016},"https:\u002F\u002Fthreejs.org","Three.js",", p5.js, and shaders for intuition about systems and state, paired with IonQ's docs, IBM Quantum Learning, Qiskit, and PennyLane, and to pick one idea, like measurement or entanglement, and build a small visual experiment around it.",[72443,73020,73021],{"avatar":104,"name":72459,"role":72460,"username":104},[18,73022,73023],{},"You do not always have to enter a field through its most technical door. Quantum became less intimidating when I stopped thinking of it only as equations and started seeing it as a way to think about invisible relationships, how separate parts of a body, or even separate people, can still affect one another.",[13,73025,73027],{"id":73026},"make-it-yours","Make it yours",[18,73029,73030],{},"If you want to see entanglement as something a body does rather than something an equation describes, Entangled Body is a good place to start. It is open for forking on Qollab and GitHub, with a live experience you can move through right now.",[73026,73032,73035],{"fork-href":72510,"live-href":73033,"title":73034},"https:\u002F\u002Fentangledbody.com","See the body that only exists in relation.",[18,73036,73037,73038],{},"Fork Entangled Body, map your own regions to qubits, and let real measurements shape the figure. ",[154,73039,73040],{},"Everything here is open and yours to build on.",[953,73042,73043],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":73045},[73046,73047,73048,73049,73050],{"id":72452,"depth":105,"text":72453},{"id":72469,"depth":105,"text":72470},{"id":72494,"depth":105,"text":72495},{"id":73001,"depth":105,"text":73002},{"id":73026,"depth":105,"text":73027},[112,969,73052],"Entangled Body",[73054,73063],{"name":72459,"role":73055,"avatar":104,"bio":73056,"links":73057},"Project lead · University of Hong Kong","A computer-engineering student at the University of Hong Kong working in frontend systems, real-time rendering, and cloud infrastructure. On Entangled Body he led the interactive layer, the point-cloud visualization, the interaction design, and how the body behaves. He came to quantum through a collaboration and a personal memory, not a physics course.",[73058,73061],{"label":73059,"href":73060},"LinkedIn ↗","https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fchanhyuk-park77177",{"label":979,"href":73062},"https:\u002F\u002Fgithub.com\u002FStree0408",{"username":72430,"name":72445,"role":73064,"avatar":104,"bio":73065,"links":73066},"Developer · National University of Singapore","A computer-science student at the National University of Singapore focused on quantum computing and algorithm design, with QKD prototypes (BB84, E91) on GitHub. On Entangled Body he built the system architecture and the generative logic that shapes the body's structure, relationships, and state transitions. He likes turning abstract physics into interactive systems.",[73067,73070,73072],{"label":73068,"href":73069},"Qollab ↗","https:\u002F\u002Fqollab.xyz\u002Fu\u002Flukeshim",{"label":73059,"href":73071},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Flukeshim030408\u002F",{"label":979,"href":73073},"https:\u002F\u002Fgithub.com\u002Flukeshim03",{"username":8,"name":73075,"role":73076,"avatar":73077},"Nicolaas Spijker","Community manager, Qollab","\u002F_content\u002Fimages\u002Fbuilders\u002Fnicolaas-spijker.webp","Chanhyuk Park and Luke Shim built a point-cloud body whose parts respond to each other across distance, entanglement you can see.","Chanhyuk Park and Luke Shim built a point-cloud human figure whose parts respond across distance: entanglement you can see. A Qollab Creative Challenge project.","Quantum Creative Challenge · Spring 2026",{"href":72510,"label":73082},"Fork Entangled Body",{"image":72502,"alt":73084,"liveUrl":73033},"Entangled Body: a point-cloud astronaut on the moon, its right arm node mapped to a 14-qubit RY→RZZ→measure circuit running on the IonQ simulator","news",{},"\u002Fblog\u002Fentangled-body","2026-07-05",[],[73091,73093,73100],{"username":1011,"project":1012,"title":1013,"category":1007,"thumb":1014,"to":73092},"\u002Fexplore\u002Fquantum-butterfly-field",{"username":73094,"project":73095,"title":73096,"category":73097,"thumb":73098,"to":73099},"incomputable","francisco","Superposition Sequencer","Music","\u002F_content\u002Fimages\u002Fsuperposition-sequencer\u002Fthumbnail.webp","\u002Fexplore\u002Ffrancisco",{"username":73101,"project":73102,"title":73103,"category":73097,"thumb":73104,"to":73105},"doraking","musiq","Musiq","\u002F_content\u002Fimages\u002Fmusiq\u002Fthumbnail.webp","\u002Fexplore\u002Fmusiq",{"title":73107,"description":73108},"Quantum Creative Project Showcase: Entangled Body","A point-cloud body whose parts respond to each other across distance. Built by Chanhyuk Park and Luke Shim for Qollab's Creative Challenge.","blog\u002Fentangled-body",[73111,143,73112],"art","generative","MpswT--5mqPtg8ztSuX1CBmHixX6LUztiQZCJQKzG2A",{"id":73115,"title":73116,"authors":73117,"body":73118,"breadcrumb":73719,"builders":73720,"byline":73731,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":73732,"description":73733,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":73734,"hero":73736,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":73737,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":73738,"publishDate":73088,"readingTime":993,"related":73739,"relatedProjects":73740,"seo":73753,"stem":73756,"tags":73757,"track":116,"trackName":116,"__hash__":73760},"blog\u002Fblog\u002Ffrancisco.md","Project Showcase: Superposition Sequencer",[73094],{"type":10,"value":73119,"toc":73712},[73120,73123,73126,73129,73137,73141,73144,73147,73152,73155,73160,73163,73167,73170,73173,73178,73181,73184,73189,73192,73654,73658,73661,73664,73669,73672,73676,73679,73682,73687,73690,73695,73697,73700,73709],[18,73121,73122],{},"Superposition Sequencer is a web-based quantum music tool that turns superposition and entanglement into something you can hear.",[18,73124,73125],{},"It looks like a normal sequencer, with drums, percussion, chords, and melodies. But the notes are not programmed directly. They are generated by running quantum circuits live on IonQ hardware, with a simulator as a fallback. You design a circuit in a visual editor, and its measured output drives the pattern: pitch, rhythm, velocity, timbre.",[18,73127,73128],{},"It is a featured project from Qollab's Spring Creative Challenge, and one of the clearest answers yet to a question the challenge keeps asking: what does quantum computing feel like when it stops being a science experiment and becomes a creative tool?",[72443,73130,73134],{"avatar":73131,"name":73132,"role":73133,"username":73094},"\u002F_content\u002Fimages\u002Fbuilders\u002Ffrancisco-estivallet.webp","Francisco Estivallet","Creator, Superposition Sequencer",[18,73135,73136],{},"I don't work with quantum regularly, but every few years I go back to it, study it a little, and get my mind blown. Then I forget about it, come back, and it happens again.",[13,73138,73140],{"id":73139},"an-instrument-not-an-experiment","An instrument, not an experiment",[18,73142,73143],{},"Francisco's way in wasn't physics, it was history. The early days of computing and electronics, he points out, came with a wave of musical exploration that opened entirely new ways to make sound, with outsized cultural influence. He thinks quantum is in a similar moment, and that the way to meet it is to make it less intimidating, which is why, of all his quantum-and-music ideas, a sequencer felt like the most approachable first step.",[18,73145,73146],{},"Most quantum demos are random-number generators in a costume. This one flips that: the composer builds the circuit, and the quantum hardware is the instrument. The real twist is what is actually being computed.",[72443,73148,73149],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,73150,73151],{},"We are not trying to compute the sequence. The sequence is the measurement. What we are computing is the relationship between instruments and effects. It is a systemic approach to musical sequencing.",[18,73153,73154],{},"It also leans into something most engineers spend their time trying to eliminate.",[72443,73156,73157],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,73158,73159],{},"In sound design, people are always looking for ways to make things sound a little imperfect. That little bit of noise is also a characteristic of quantum computers, so it could bring a new personality to the sound itself.",[18,73161,73162],{},"It is also built to last, not to be a one-off demo. Circuits and the patterns they generate can be exported, saved, loaded, and reused, so you build up a personal library of quantum musical ideas, much like a hardware synth's patch memory. A downloadable library of pre-executed sequences even lets musicians use the output without their own hardware access.",[13,73164,73166],{"id":73165},"how-francisco-built-it","How Francisco built it",[18,73168,73169],{},"Francisco's first prototype looked nothing like a sequencer. It was a rotating sphere whose cursor struck different notes as it turned, leaning on the geometric way we usually picture quantum states. It didn't fit the metaphors he was after, so he went back to something more familiar: a step sequencer where each track line is a single shot of the circuit.",[18,73171,73172],{},"Under the hood, Qiskit builds the circuits and each run, on IonQ or the local simulator, produces probability distributions or entangled measurement outcomes that map onto musical parameters: pitch selection, rhythm, velocity, timbre. The circuits are fully parameterized, so you can modify them without a full recompile and explore how each change reshapes the sound.",[72443,73174,73175],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,73176,73177],{},"Each line is one shot of the circuit. It's that randomness and change between shots that creates the beat.",[18,73179,73180],{},"On the audio side, Tone.js handles sample-accurate scheduling, synthesis, and effects, all fed by the quantum-generated modulations. The app is fully serverless: a SvelteKit front end (with Threlte and Three.js driving the live Bloch spheres) talks to a stateless FastAPI service running Qiskit, which builds the circuit, extracts each qubit's Bloch vector layer by layer, and samples outcomes from the full statevector. Qiskit Aer and IonQ sit behind the same API.",[18,73182,73183],{},"To keep it responsive, frequently-used patterns are precomputed and cached while rare or highly-parameterized circuits trigger fresh hardware runs, balancing hardware cost against live interactivity.",[831,73185],{"caption":73186,"no":844,"poster":73187,"video":73188},"Building a circuit in the visual editor and hearing it drive the sequencer. Press play, sound on.","\u002F_content\u002Fimages\u002Fsuperposition-sequencer\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F2b1e9e5e-f28d-4ce2-8374-c59cdb281806",[18,73190,73191],{},"The same circuit renders to music outside the browser, too. This is the gist of the script you can copy onto Qollab's Playground to turn a circuit into a MIDI file: each qubit is a track, each shot is a beat, and a separate track writes how entangled each pair of qubits is as a MIDI control signal.",[493,73193,73196],{"name":73194,"run-href":73195,"tag":496},"render_midi.py","\u002Fu\u002Fincomputable\u002Fsuperposition-sequencer",[498,73197,73199],{"className":500,"code":73198,"language":502,"meta":104,"style":104},"# 'backend' is pre-created as a global on Qollab\nfrom qiskit import QuantumCircuit\nfrom qiskit.providers.jobstatus import JobStatus\nimport time, random\n\nNUM_QUBITS = 4\nBPM = 104\nSHOTS = 104  # 104 shots ≈ one minute at 104 BPM\n\n# qubit index → instrument (each qubit is a track)\nSOUNDS = {0: \"clap\", 1: \"clap\", 2: \"bell\", 3: \"tom\"}\n\n# Build the circuit. Each layer (column of gates) is a beat.\ncircuit = QuantumCircuit(NUM_QUBITS, NUM_QUBITS)\ncircuit.h(0)\ncircuit.h(2)\ncircuit.ry(0.976411, 3)\ncircuit.cx(0, 1)\ncircuit.measure(range(NUM_QUBITS), range(NUM_QUBITS))\n\ndef main(shots=SHOTS):\n    job = backend.run(circuit, shots=shots)\n    while job.status() is not JobStatus.DONE:\n        time.sleep(10)\n    counts = job.get_counts()\n\n    # Each shot is one beat. Rebuild an ordered shot list from the counts,\n    # then shuffle so identical outcomes aren't clumped at the start.\n    shots = []\n    for bits, c in counts.items():\n        shots.extend([bits.replace(\" \", \"\")] * c)\n    random.shuffle(shots)\n    write_midi(shots)\n",[504,73200,73201,73206,73216,73228,73235,73239,73249,73259,73272,73276,73281,73326,73330,73335,73354,73375,73387,73409,73433,73457,73461,73478,73509,73533,73547,73561,73565,73570,73575,73584,73600,73630,73641],{"__ignoreMap":104},[507,73202,73203],{"class":509,"line":510},[507,73204,73205],{"class":562},"# 'backend' is pre-created as a global on Qollab\n",[507,73207,73208,73210,73212,73214],{"class":509,"line":105},[507,73209,529],{"class":513},[507,73211,532],{"class":517},[507,73213,514],{"class":513},[507,73215,537],{"class":517},[507,73217,73218,73220,73223,73225],{"class":509,"line":540},[507,73219,529],{"class":513},[507,73221,73222],{"class":517}," qiskit.providers.jobstatus ",[507,73224,514],{"class":513},[507,73226,73227],{"class":517}," JobStatus\n",[507,73229,73230,73232],{"class":509,"line":553},[507,73231,514],{"class":513},[507,73233,73234],{"class":517}," time, random\n",[507,73236,73237],{"class":509,"line":559},[507,73238,556],{"emptyLinePlaceholder":133},[507,73240,73241,73244,73246],{"class":509,"line":566},[507,73242,73243],{"class":583},"NUM_QUBITS",[507,73245,1423],{"class":572},[507,73247,73248],{"class":583}," 4\n",[507,73250,73251,73254,73256],{"class":509,"line":590},[507,73252,73253],{"class":583},"BPM",[507,73255,1423],{"class":572},[507,73257,73258],{"class":583}," 104\n",[507,73260,73261,73264,73266,73269],{"class":509,"line":610},[507,73262,73263],{"class":583},"SHOTS",[507,73265,1423],{"class":572},[507,73267,73268],{"class":583}," 104",[507,73270,73271],{"class":562},"  # 104 shots ≈ one minute at 104 BPM\n",[507,73273,73274],{"class":509,"line":634},[507,73275,556],{"emptyLinePlaceholder":133},[507,73277,73278],{"class":509,"line":661},[507,73279,73280],{"class":562},"# qubit index → instrument (each qubit is a track)\n",[507,73282,73283,73286,73288,73291,73293,73295,73298,73300,73302,73304,73306,73308,73310,73312,73315,73317,73319,73321,73324],{"class":509,"line":678},[507,73284,73285],{"class":583},"SOUNDS",[507,73287,1423],{"class":572},[507,73289,73290],{"class":517}," {",[507,73292,601],{"class":583},[507,73294,1403],{"class":517},[507,73296,73297],{"class":730},"\"clap\"",[507,73299,622],{"class":517},[507,73301,625],{"class":583},[507,73303,1403],{"class":517},[507,73305,73297],{"class":730},[507,73307,622],{"class":517},[507,73309,584],{"class":583},[507,73311,1403],{"class":517},[507,73313,73314],{"class":730},"\"bell\"",[507,73316,622],{"class":517},[507,73318,8226],{"class":583},[507,73320,1403],{"class":517},[507,73322,73323],{"class":730},"\"tom\"",[507,73325,23875],{"class":517},[507,73327,73328],{"class":509,"line":683},[507,73329,556],{"emptyLinePlaceholder":133},[507,73331,73332],{"class":509,"line":697},[507,73333,73334],{"class":562},"# Build the circuit. Each layer (column of gates) is a beat.\n",[507,73336,73337,73340,73342,73344,73346,73348,73350,73352],{"class":509,"line":710},[507,73338,73339],{"class":517},"circuit ",[507,73341,573],{"class":572},[507,73343,577],{"class":576},[507,73345,580],{"class":517},[507,73347,73243],{"class":583},[507,73349,622],{"class":517},[507,73351,73243],{"class":583},[507,73353,587],{"class":517},[507,73355,73356,73359,73361,73363,73365,73367],{"class":509,"line":715},[507,73357,73358],{"class":517},"circuit.",[507,73360,596],{"class":576},[507,73362,580],{"class":517},[507,73364,601],{"class":583},[507,73366,3649],{"class":517},[507,73368,72708,73369],{"class":72706,"tabindex":72707},[507,73370,73371,73374],{"class":72711,"role":72712},[154,73372,73373],{},"Superposition."," About a 50% chance this track fires on any given beat.",[507,73376,73377,73379,73381,73383,73385],{"class":509,"line":721},[507,73378,73358],{"class":517},[507,73380,596],{"class":576},[507,73382,580],{"class":517},[507,73384,584],{"class":583},[507,73386,587],{"class":517},[507,73388,73389,73391,73393,73395,73398,73400,73402,73404],{"class":509,"line":736},[507,73390,73358],{"class":517},[507,73392,639],{"class":576},[507,73394,580],{"class":517},[507,73396,73397],{"class":583},"0.976411",[507,73399,622],{"class":517},[507,73401,8226],{"class":583},[507,73403,3649],{"class":517},[507,73405,72708,73406],{"class":72706,"tabindex":72707},[507,73407,73408],{"class":72711,"role":72712},"A rotation tunes how often this track hits: the \"how often\" knob.",[507,73410,73411,73413,73415,73417,73419,73421,73423,73425],{"class":509,"line":748},[507,73412,73358],{"class":517},[507,73414,615],{"class":576},[507,73416,580],{"class":517},[507,73418,601],{"class":583},[507,73420,622],{"class":517},[507,73422,625],{"class":583},[507,73424,3649],{"class":517},[507,73426,72708,73427],{"class":72706,"tabindex":72707},[507,73428,73429,73432],{"class":72711,"role":72712},[154,73430,73431],{},"Entanglement."," Tracks 0 and 1 fire together, though each one alone still looks like a coin flip.",[507,73434,73435,73437,73439,73441,73443,73445,73447,73449,73451,73453,73455],{"class":509,"line":761},[507,73436,73358],{"class":517},[507,73438,72822],{"class":576},[507,73440,580],{"class":517},[507,73442,2204],{"class":572},[507,73444,580],{"class":517},[507,73446,73243],{"class":583},[507,73448,2213],{"class":517},[507,73450,2204],{"class":572},[507,73452,580],{"class":517},[507,73454,73243],{"class":583},[507,73456,22540],{"class":517},[507,73458,73459],{"class":509,"line":775},[507,73460,556],{"emptyLinePlaceholder":133},[507,73462,73463,73465,73468,73470,73472,73474,73476],{"class":509,"line":784},[507,73464,1370],{"class":513},[507,73466,73467],{"class":576}," main",[507,73469,580],{"class":517},[507,73471,68762],{"class":1382},[507,73473,573],{"class":517},[507,73475,73263],{"class":583},[507,73477,1883],{"class":517},[507,73479,73480,73483,73485,73488,73490,73493,73495,73497,73500],{"class":509,"line":796},[507,73481,73482],{"class":517},"    job ",[507,73484,573],{"class":572},[507,73486,73487],{"class":517}," backend.",[507,73489,22501],{"class":576},[507,73491,73492],{"class":517},"(circuit, ",[507,73494,68762],{"class":2155},[507,73496,573],{"class":572},[507,73498,73499],{"class":517},"shots)",[507,73501,72708,73502],{"class":72706,"tabindex":72707},[507,73503,73504,73505,73508],{"class":72711,"role":72712},"Runs on a trapped-ion IonQ machine through Qollab. ",[504,73506,73507],{},"backend"," is provided for you.",[507,73510,73511,73514,73516,73519,73521,73523,73525,73528,73531],{"class":509,"line":809},[507,73512,73513],{"class":513},"    while",[507,73515,23993],{"class":517},[507,73517,73518],{"class":576},"status",[507,73520,1677],{"class":517},[507,73522,37008],{"class":513},[507,73524,21980],{"class":513},[507,73526,73527],{"class":517}," JobStatus.",[507,73529,73530],{"class":583},"DONE",[507,73532,1728],{"class":517},[507,73534,73535,73538,73541,73543,73545],{"class":509,"line":1352},[507,73536,73537],{"class":517},"        time.",[507,73539,73540],{"class":576},"sleep",[507,73542,580],{"class":517},[507,73544,23805],{"class":583},[507,73546,587],{"class":517},[507,73548,73549,73552,73554,73556,73559],{"class":509,"line":1357},[507,73550,73551],{"class":517},"    counts ",[507,73553,573],{"class":572},[507,73555,23993],{"class":517},[507,73557,73558],{"class":576},"get_counts",[507,73560,781],{"class":517},[507,73562,73563],{"class":509,"line":1362},[507,73564,556],{"emptyLinePlaceholder":133},[507,73566,73567],{"class":509,"line":1367},[507,73568,73569],{"class":562},"    # Each shot is one beat. Rebuild an ordered shot list from the counts,\n",[507,73571,73572],{"class":509,"line":1379},[507,73573,73574],{"class":562},"    # then shuffle so identical outcomes aren't clumped at the start.\n",[507,73576,73577,73580,73582],{"class":509,"line":1389},[507,73578,73579],{"class":517},"    shots ",[507,73581,573],{"class":572},[507,73583,1910],{"class":517},[507,73585,73586,73588,73591,73593,73596,73598],{"class":509,"line":1397},[507,73587,1916],{"class":513},[507,73589,73590],{"class":517}," bits, c ",[507,73592,1636],{"class":513},[507,73594,73595],{"class":517}," counts.",[507,73597,22607],{"class":576},[507,73599,1930],{"class":517},[507,73601,73602,73605,73608,73611,73613,73615,73618,73620,73622,73625,73627],{"class":509,"line":1412},[507,73603,73604],{"class":517},"        shots.",[507,73606,73607],{"class":576},"extend",[507,73609,73610],{"class":517},"([bits.",[507,73612,69532],{"class":576},[507,73614,580],{"class":517},[507,73616,73617],{"class":730},"\" \"",[507,73619,622],{"class":517},[507,73621,8430],{"class":730},[507,73623,73624],{"class":517},")] ",[507,73626,2391],{"class":572},[507,73628,73629],{"class":517}," c)\n",[507,73631,73632,73635,73638],{"class":509,"line":1431},[507,73633,73634],{"class":517},"    random.",[507,73636,73637],{"class":576},"shuffle",[507,73639,73640],{"class":517},"(shots)\n",[507,73642,73643,73646,73649],{"class":509,"line":1449},[507,73644,73645],{"class":576},"    write_midi",[507,73647,73648],{"class":517},"(shots)",[507,73650,72708,73651],{"class":72706,"tabindex":72707},[507,73652,73653],{"class":72711,"role":72712},"Writes one MIDI track per qubit, plus an entanglement track where each pair's mutual information becomes a MIDI control signal.",[13,73655,73657],{"id":73656},"learning-by-ear","Learning by ear",[18,73659,73660],{},"Beyond making music, the project is a way into quantum thinking. Concepts like superposition, measurement, and entanglement become tangible when they are mapped to something you can hear: experiment with a circuit, watch the probability distribution shift, and listen to the pattern change with it.",[18,73662,73663],{},"Francisco's hope is specific. Just as synthesizers once pulled artists into electronics and then software in search of new sounds, building real intuition along the way, he wants this to do the same for quantum.",[72443,73665,73666],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,73667,73668],{},"Even if you don't understand a thing about quantum, you get a good sense of what a circuit does to the outputs, hopefully in a less intimidating way, so you feel comfortable diving into the rabbit hole.",[18,73670,73671],{},"The whole thing is open source, including the circuits, the front-end code, example projects, and documentation. Developers can extend it into new domains, artists can remix it, and educators can adapt it for workshops, lowering the barrier to quantum while showing a genuinely new way to interact with it.",[13,73673,73675],{"id":73674},"where-its-headed","Where it's headed",[18,73677,73678],{},"Francisco has a long list of where this goes: richer preset circuits built with quantum researchers, richer compositions and sound design with musicians, and a physical version you could play like a real instrument, something he is already eyeing for festivals. On the software side, he wants external signal inputs, a plugin for music software, and quantum-generated scales, chords, and instruments.",[18,73680,73681],{},"The nearest piece is the sound design itself, entanglement especially.",[72443,73683,73684],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,73685,73686],{},"When I have entanglement it's not doing anything yet, and I want to play with that, how to mix the two sounds so they represent that connection. It'll be more of an ambient sound that shifts as you get closer to the state.",[18,73688,73689],{},"Underneath the feature list is the conviction that keeps pulling him back to quantum in the first place.",[72443,73691,73692],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,73693,73694],{},"Now more than ever, people are tuned to how technologies can influence our lives quicker than we're ready for. Becoming familiar with them is an important tool in adapting.",[13,73696,73027],{"id":73026},[18,73698,73699],{},"If you want to hear what a quantum circuit sounds like, this is a good place to start. Superposition Sequencer is open for forking on both Qollab and GitHub, and there is a live demo you can play with right now.",[73026,73701,73704],{"fork-href":73195,"live-href":73702,"title":73703},"https:\u002F\u002Fsuperposition-sequencer.incomputable.io","Play a quantum circuit.",[18,73705,73706,73707],{},"Fork the sequencer, build a circuit, and hear it run on real hardware. ",[154,73708,73040],{},[953,73710,73711],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":73713},[73714,73715,73716,73717,73718],{"id":73139,"depth":105,"text":73140},{"id":73165,"depth":105,"text":73166},{"id":73656,"depth":105,"text":73657},{"id":73674,"depth":105,"text":73675},{"id":73026,"depth":105,"text":73027},[112,969,73096],[73721],{"username":73094,"name":73132,"role":73722,"avatar":73131,"bio":73723,"links":73724},"Creative technologist · Incomputable","Francisco is a mechatronics engineer and creative technologist with 15+ years across robotics, industrial automation, data products, and interactive installations. Through his Barcelona practice Incomputable, he builds tools and prototypes for artists, designers, and research labs, recently with six installations at Dubai's Sikka Art Festival and a prototype for the EU S+T+Arts residency. He is faculty at IAAC and came to quantum from the creative-coding and sonification side rather than physics.",[73725,73727,73729],{"label":73068,"href":73726},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fincomputable",{"label":73059,"href":73728},"https:\u002F\u002Flinkedin.com\u002Fin\u002Ffranciscoestivallet",{"label":979,"href":73730},"https:\u002F\u002Fgithub.com\u002Fchicoe",{"username":8,"name":73075,"role":73076,"avatar":73077},"Francisco Estivallet built a music sequencer where every note comes from a quantum circuit you design, turning a quantum computer into a playable instrument.","Francisco Estivallet built a quantum music sequencer where every note comes from a circuit you design. A Qollab Creative Challenge project.",{"href":73195,"label":73735},"Fork the sequencer",{"image":104,"alt":104,"liveUrl":73702},{},"\u002Fblog\u002Ffrancisco",[],[73741,73747,73751],{"username":73742,"project":73743,"title":73744,"category":73097,"thumb":73745,"to":73746},"cephasteom","quantum-patterns","Quantum Patterns","\u002F_content\u002Fimages\u002Fquantum-patterns\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-patterns",{"username":72430,"project":73748,"title":73052,"category":1007,"thumb":73749,"to":73750},"entangled-body","\u002F_content\u002Fimages\u002Fentangled-body\u002Fthumbnail.webp","\u002Fexplore\u002Fentangled-body",{"username":997,"project":998,"title":999,"category":73752,"thumb":1001,"to":1002},"Education",{"title":73754,"description":73755},"Quantum Creative Project Showcase: Superposition Sequencer","A quantum music sequencer where the notes come from circuits you build, run on real IonQ hardware. Built by Francisco Estivallet for Qollab's Creative Challenge.","blog\u002Ffrancisco",[73758,143,73759],"music","creative-coding","07uzHUl3vDOjIPJV6ExHAJTqhebUGkKAVMe8N2LSYuc",{"id":73762,"title":73763,"authors":73764,"body":73765,"breadcrumb":73769,"builders":73771,"byline":116,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":73772,"description":73773,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":73774,"navigation":133,"newsItems":73775,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":73798,"publishDate":73088,"readingTime":116,"related":73799,"relatedProjects":116,"seo":73800,"stem":73803,"tags":73804,"track":116,"trackName":116,"__hash__":73805},"blog\u002Fblog\u002Fnews.md","News & open calls",[8],{"type":10,"value":73766,"toc":73767},[],{"title":104,"searchDepth":105,"depth":105,"links":73768},[],[112,969,73770],"News",[],"Open calls, programs, and what's new on Qollab. Get funded, get involved, and keep up with the platform.","Open calls, programs, and product updates from Qollab. Apply to the Creative Challenge, become an ambassador, and see what's new on the platform.",{},[73776,73781,73787,73793],{"title":72285,"desc":73777,"date":73778,"href":73779,"cta":73780},"Playground upgrades, new features, and shipped improvements. The running changelog.","June 24, 2026","\u002Fexplore\u002Fchangelog","See the changelog",{"title":73782,"desc":73783,"date":73784,"href":73785,"cta":73786},"Meet the Spring 2026 cohort","13 teams building on real IonQ hardware: tools, music, visualizations, and games. The lineup, project by project.","May 5, 2026","\u002Fexplore\u002Fspring-2026-cohort","Read the roundup",{"title":73788,"desc":73789,"date":73790,"href":73791,"cta":73792},"Become a Qollab Ambassador","Champion quantum in your community and earn monthly IonQ hardware credits.","May 1, 2026","\u002Fexplore\u002Fambassadors","Learn more",{"title":73794,"desc":73795,"date":73796,"href":73797,"cta":73792},"Creative Challenge: submissions closed","Grants and IonQ compute credits for open-source quantum projects. The call is closed — see what the funded teams built.","April 7, 2026","\u002Fexplore\u002Fcreative-challenge","\u002Fblog\u002Fnews",[],{"title":73801,"description":73802},"What's Happening on Qollab","Open calls, programs, and product updates from the quantum developer community.","blog\u002Fnews",[73085,143],"ywWu4bavjmb4MM0IqTIBpFLncwUYDhDj2AYOqeJzI-E",{"id":73807,"title":73808,"authors":73809,"body":73810,"breadcrumb":74123,"builders":74124,"byline":74125,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":74126,"description":74127,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":74128,"lessonCount":116,"meta":74129,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":74130,"publishDate":74131,"readingTime":74132,"related":74133,"relatedProjects":116,"seo":74134,"stem":74137,"tags":74138,"track":116,"trackName":116,"__hash__":74139},"blog\u002Fblog\u002Fart.md","Can a Quantum Computer Make Art?",[8],{"type":10,"value":73811,"toc":74112},[73812,73815,73818,73825,73829,73832,73835,73839,73842,73913,73917,73920,73932,73939,73942,73946,73950,73953,73959,73977,73985,73988,73995,73999,74002,74006,74009,74014,74017,74021,74025,74028,74031,74036,74042,74046,74049,74052,74055,74059,74064,74068,74071,74074,74077,74081,74084,74087,74091,74095,74098],[18,73813,73814],{},"A quantum computer does not draw pictures. Its raw output is a probability distribution, a spread of numbers with no obvious shape. Turning that spread into something you can see and feel is one of the more direct ways to make quantum behavior real to a person, instead of a proof on a page.",[18,73816,73817],{},"This is a story about three artworks that do exactly that. They came out of Qollab's Quantum Creative Challenge, and each one takes the strangest idea in quantum physics, entanglement, and grows it inside something living: a field of butterflies, a garden, and a human body. None of them asks you to know any physics first. That is the whole point.",[72443,73819,73822],{"avatar":104,"name":73820,"role":73821,"username":104},"Carlo Rovelli","Helgoland",[18,73823,73824],{},"You look at a butterfly and see the color of its wings. In relation to me, a relation is established between you and the butterfly: the butterfly and you are now in an entangled state. Everything in the world does not exist other than in this web of entanglement.",[13,73826,73828],{"id":73827},"what-is-quantum-art","What is quantum art?",[18,73830,73831],{},"Quantum art, in the sense that matters here, is art made with a real quantum computer, where the machine's actual behavior, its measurements, probabilities, and entanglement, becomes the raw material. That is different from art that only borrows the word, or uses a quantum gadget as a fancy random-number generator.",[18,73833,73834],{},"It is a small but real field, and 2025 gave it a spotlight when the United Nations named it the International Year of Quantum Science and Technology. What is new on Qollab is not the idea but the access. Three teams from its Creative Challenge built three artworks you can open in a browser, run on real hardware, and take apart: butterflies that heal after damage, a garden that grows from measurement, and a body whose parts answer each other across distance.",[13,73836,73838],{"id":73837},"at-a-glance","At a glance",[18,73840,73841],{},"Three teams, three living forms, one shared idea: let entanglement happen somewhere you can watch it. Here is who built what.",[41852,73843,73844,73863],{},[41855,73845,73846],{},[41858,73847,73848,73851,73854,73857,73860],{},[41861,73849,73850],{},"Project",[41861,73852,73853],{},"The idea",[41861,73855,73856],{},"Runs on",[41861,73858,73859],{},"What you see",[41861,73861,73862],{},"Built by",[41868,73864,73865,73881,73897],{},[41858,73866,73867,73869,73872,73875,73878],{},[41873,73868,1013],{},[41873,73870,73871],{},"Healing",[41873,73873,73874],{},"IonQ Forte hardware",[41873,73876,73877],{},"Five qubits scramble into one field; one is damaged, then recovered",[41873,73879,73880],{},"Xinyi Zhang",[41858,73882,73883,73885,73888,73891,73894],{},[41873,73884,1006],{},[41873,73886,73887],{},"Growth",[41873,73889,73890],{},"IonQ hardware, from a pool of runs",[41873,73892,73893],{},"Plants whose form is fixed by real quantum measurements",[41873,73895,73896],{},"Amber Wang & Justin Pincar",[41858,73898,73899,73901,73904,73907,73910],{},[41873,73900,73052],{},[41873,73902,73903],{},"Connection",[41873,73905,73906],{},"IonQ hardware (added later)",[41873,73908,73909],{},"A point-cloud body whose parts respond across distance",[41873,73911,73912],{},"Chanhyuk Park & Luke Shim",[13,73914,73916],{"id":73915},"why-something-living","Why something living?",[18,73918,73919],{},"Entanglement is famously hard to picture. Two particles can share a single state so completely that neither has one of its own, and measuring one tells you about the other, however far apart they sit. Equations capture that precisely. A living thing captures it differently.",[18,73921,73922,73923,73927,73928,73931],{},"All three artworks make the same bet: that a butterfly, a plant, or a body is easier to feel than a state vector. It is not a new bet. The artist ",[49,73924,73926],{"href":73925},"https:\u002F\u002Fscheringstiftung.de\u002Fen\u002Fprojektraum\u002Flibby-heaney\u002F","Libby Heaney"," has staged entanglement inside a version of Bosch's ",[1031,73929,73930],{},"Garden of Earthly Delights",", and the sculptor Julian Voss-Andreae has put quantum states into the human figure for years. What these three add is that the living form is driven by a real circuit, not just illustrated by one.",[72443,73933,73936],{"avatar":73934,"name":73880,"role":73935,"username":1011},"\u002F_content\u002Fimages\u002Fbuilders\u002Fxinyi-zhang.webp","Creator, Quantum Butterfly Field",[18,73937,73938],{},"This quantum resilience resonates with the Native Hawaiian concept of lōkahi, unity and wholeness, where individual wellbeing is maintained through the integrity of our relationships within a web of the interconnected whole.",[18,73940,73941],{},"Chanhyuk Park, who built Entangled Body, arrived at the same place from the body's side rather than the physics.",[72443,73943,73944],{"avatar":104,"name":72459,"role":72460,"username":104},[18,73945,72478],{},[13,73947,73949],{"id":73948},"the-opportunity","The opportunity",[18,73951,73952],{},"For most of its history, making art with a quantum computer meant institutional access: a lab, a university, a research partnership. The machines lived behind those doors.",[18,73954,73955,73956,73958],{},"That is what changed. Cloud quantum computers you can reach from a browser tab are recent, and ",[154,73957,112],{}," is a community and coding platform built around that shift: a place to write quantum code and run it on IonQ's trapped-ion hardware, with no lab or affiliation required. In spring 2026, Qollab and IonQ funded a Creative Challenge on top of it, compute credits, cash, and mentorship for open, original projects from anyone with an idea. Quantum Butterfly Field and Entangled Body came out of that round, and Quantum Garden from the one before. All three are open source, and none was built only by physicists.",[18,73960,73961,73962,73966,73967,73971,73972,73976],{},"The timing is not an accident, because the wider art world is paying attention too. In 2025, Science Gallery London opened a six-month show called ",[49,73963,73965],{"href":73964},"https:\u002F\u002Flondon.sciencegallery.com\u002Fquantum","Quantum Untangled",", Tokyo's Museum of Contemporary Art exhibited ",[49,73968,73970],{"href":73969},"https:\u002F\u002Fwww.mot-art-museum.jp\u002Fen\u002Fexhibitions\u002Fmission-infinity\u002F","what it billed as the first artwork made with a Japanese quantum computer",", and the LAS Art Foundation's ",[49,73973,73975],{"href":73974},"https:\u002F\u002Fwww.las-art.foundation\u002Fexplore\u002Fsensing-quantum","Sensing Quantum"," program won a Grand Prize at the EU's S+T+ARTS Awards. The honest question running under all of it is when a work is really quantum and when the word is just decoration. These three answer it by showing their circuits.",[72443,73978,73982],{"avatar":73979,"name":73980,"role":73981,"username":1004},"\u002F_content\u002Fimages\u002Fbuilders\u002Famber-wang.webp","Amber Wang","Co-creator, Quantum Garden",[18,73983,73984],{},"You don't need to be a physicist to do something meaningful with quantum; you need a strong concept about time, uncertainty, or connection, and a willingness to collaborate with technical partners. Think of quantum as a new storytelling and interaction medium, not just a buzzword or a black box.",[18,73986,73987],{},"Justin Pincar, who built Quantum Garden with her, spent years thinking quantum was out of reach. What changed his mind was simply being able to reach it.",[72443,73989,73992],{"avatar":73990,"name":73991,"role":73981,"username":1004},"\u002F_content\u002Fimages\u002Fbuilders\u002Fjustin-pincar.webp","Justin Pincar",[18,73993,73994],{},"I realized that through platforms like Qollab and IonQ, you can access real quantum hardware as simply as spinning up any other cloud service. That was the moment it clicked and I realized that it was actually accessible now, not just theoretical.",[13,73996,73998],{"id":73997},"where-this-goes","Where this goes",[18,74000,74001],{},"What exists today is a first pass. Each piece maps one circuit to one experience. The builders are already thinking bigger.",[72443,74003,74004],{"avatar":104,"name":72445,"role":72446,"username":72430},[18,74005,73010],{},[18,74007,74008],{},"For Amber, the direction is to treat the quantum behavior itself as the medium, the way a painter treats paint.",[72443,74010,74011],{"avatar":73979,"name":73980,"role":73981,"username":1004},[18,74012,74013],{},"Designing for quantum means treating concepts like superposition, entanglement, and probabilistic measurement as actual creative materials, not just technical details. Instead of thinking, \"What output do I want?\" you're asking, \"What distribution of possible states do I want, and how should people encounter those states over time?\"",[18,74015,74016],{},"The larger bet is the one generative art and creative coding made before: a new material pulls in people who would never open a physics textbook, and some of them stay long enough to learn what is underneath. If quantum art follows that path, the lasting effect may not be the artworks at all. It may be who ends up curious about qubits because a butterfly, a garden, or a body got to them first.",[13,74018,74020],{"id":74019},"quantum-butterfly-field-five-qubits-one-field","Quantum Butterfly Field: five qubits, one field",[74022,74023],"chapter-meta",{"fork":74024,"showcase":73092,"who":73880},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fxinyi\u002Fquantum-butterfly-field",[18,74026,74027],{},"Quantum Butterfly Field is an interactive artwork by Xinyi Zhang, an artist and technologist with computer-science degrees from MIT and the University of British Columbia. Five butterflies are five qubits. As a circuit runs, it scrambles their identities into one entangled field, until no single butterfly holds its own state anymore. Then one is damaged and cut off, and in a classical world that loss would be permanent.",[18,74029,74030],{},"It is not. Through a real result from quantum information theory that Zhang stages as the anti-butterfly effect, the lost butterfly reassembles from the correlations the others still hold. Everything you see, the sharpness of a wing, the threads drawn between them, the glow of the healed one, is driven by a real quantum measure computed as the circuit runs. No numbers ever appear on screen.",[831,74032],{"alt":74033,"caption":104,"no":104,"poster":74034,"video":74035},"Quantum Butterfly Field: five butterflies as five qubits in one entangled field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F3768811c-0ddc-4ad3-b2cd-416c8cebee9c",[72443,74037,74039],{"avatar":73934,"name":73880,"role":74038,"username":1011},"Artist statement",[18,74040,74041],{},"What does it mean to heal in a quantum world?",[13,74043,74045],{"id":74044},"quantum-garden-grown-from-measurement","Quantum Garden: grown from measurement",[74022,74047],{"fork":74048,"showcase":1009,"who":73896},"https:\u002F\u002Fqollab.xyz\u002Fu\u002FAmberPincar\u002Fquantum-garden",[18,74050,74051],{},"Quantum Garden is a living digital garden by Amber Wang, a data scientist, and Justin Pincar, a software engineer. Every plant gets its form, color, and behavior from a real quantum measurement, drawn from a pool of results run on IonQ hardware in advance. Not simulated randomness, but actual outcomes, baked permanently into each plant the first time you look at it.",[18,74053,74054],{},"Some plants are entangled with others across the garden, so observing one tells you something about a plant you have not visited yet. And the garden keeps its own time, germinating and fading whether or not anyone is watching. The hardware's slowness, once a problem, became the reason the garden has seasons.",[831,74056],{"alt":74057,"caption":104,"no":104,"src":74058},"Quantum Garden, a generative garden whose plants come from real quantum hardware","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fhero-creative-challenge.webp",[72443,74060,74061],{"avatar":73990,"name":73991,"role":73981,"username":1004},[18,74062,74063],{},"Working with quantum requires a different way of thinking. Classical programming is deterministic, and you get the output you expect. With quantum, you're dealing with probabilities and superpositions, not exactly random, but similar in practice.",[13,74065,74067],{"id":74066},"entangled-body-entanglement-you-can-touch","Entangled Body: entanglement you can touch",[74022,74069],{"fork":74070,"showcase":73750,"who":73912},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Flukeshim\u002Fentangled-body",[18,74072,74073],{},"Entangled Body, by Chanhyuk Park and Luke Shim, is a point-cloud human figure whose parts respond to each other across distance. Touch one region and somewhere else reacts; move your view and the body resolves and dissolves. Fourteen body regions map to fourteen qubits, coupled along a real anatomical graph, so a touch ripples through the figure the way entanglement ripples through a circuit.",[18,74075,74076],{},"Of the three, this one leads with the art. It began as a quantum-inspired piece and added a real circuit later, and its builders are candid about that order. They also kept the hardware's noise in rather than smoothing it away, because a body, to them, should always be becoming rather than fixed.",[72443,74078,74079],{"avatar":104,"name":72459,"role":72460,"username":104},[18,74080,72491],{},[831,74082],{"alt":74083,"caption":104,"no":104,"poster":72502,"video":72503},"Entangled Body: a point-cloud astronaut whose regions are coupled like entangled qubits",[18,74085,74086],{},"Luke Shim, who built the system underneath, treats that uncertainty as the material rather than a flaw.",[72443,74088,74089],{"avatar":104,"name":72445,"role":72446,"username":72430},[18,74090,72995],{},[13,74092,74094],{"id":74093},"try-one-yourself","Try one yourself",[18,74096,74097],{},"Every artwork here is open source, and you can run one in your browser right now. Fork the circuit behind it, change the parameters, and see what happens on real hardware. Start with whichever one pulled you in.",[74099,74100,74107],"fork-row",{"c1":74101,"c2":74102,"c3":74103,"f1":74024,"f2":74048,"f3":74070,"l1":74104,"l2":74105,"l3":73082,"title":74106},"#ff78b6","#e254c6","#9660f0","Fork Butterfly Field","Fork Quantum Garden","Make your own.",[18,74108,74109,74110],{},"Fork any of the three, run its circuit on real IonQ hardware, and make it yours. ",[154,74111,73040],{},{"title":104,"searchDepth":105,"depth":105,"links":74113},[74114,74115,74116,74117,74118,74119,74120,74121,74122],{"id":73827,"depth":105,"text":73828},{"id":73837,"depth":105,"text":73838},{"id":73915,"depth":105,"text":73916},{"id":73948,"depth":105,"text":73949},{"id":73997,"depth":105,"text":73998},{"id":74019,"depth":105,"text":74020},{"id":74044,"depth":105,"text":74045},{"id":74066,"depth":105,"text":74067},{"id":74093,"depth":105,"text":74094},[112,969,1007],[],{"username":8,"name":73075,"role":73076,"avatar":73077},"Three teams from Qollab's Creative Challenge answered yes, on real IonQ hardware: a field of butterflies that heal after damage, a garden grown from quantum measurements, and a human body whose parts respond across distance. Here is what quantum art is, in the work of the people making it.","Three teams from Qollab made art on real IonQ quantum hardware: healing butterflies, a garden grown from measurement, a body you can touch. Open and forkable.","topic",{},"\u002Fblog\u002Fart","2026-07-04","8 min read",[],{"title":74135,"description":74136},"Art Made With a Quantum Computer","Three teams from Qollab's Quantum Creative Challenge made art on real IonQ quantum hardware. Open source and forkable.","blog\u002Fart",[73111,143],"wbkgyvFFoAKqaszVffz8VNjGkXEjSfa9RGopdFfISBw",{"id":74141,"title":74142,"authors":74143,"body":74144,"breadcrumb":74464,"builders":74465,"byline":74466,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":74467,"description":74468,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":74128,"lessonCount":116,"meta":74469,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":74470,"publishDate":74131,"readingTime":74132,"related":74471,"relatedProjects":116,"seo":74472,"stem":74475,"tags":74476,"track":116,"trackName":116,"__hash__":74478},"blog\u002Fblog\u002Feducation.md","See a Quantum Computer Actually Work",[8],{"type":10,"value":74145,"toc":74451},[74146,74149,74152,74156,74159,74166,74173,74177,74186,74188,74306,74310,74313,74316,74322,74326,74330,74333,74340,74344,74348,74351,74358,74362,74366,74369,74376,74380,74384,74387,74394,74398,74402,74405,74413,74417,74425,74428,74436,74438,74441],[18,74147,74148],{},"A quantum computer's inner life is invisible. The thing doing the actual work, a state spread across many possibilities at once, never shows up on a screen. You are usually asked to take it on faith, or on math.",[18,74150,74151],{},"The six projects on this page refuse that. Each one takes something you normally cannot see, an algorithm running, a molecule forming, a quantum computer racing an ordinary one, and turns it into something on screen you can poke at. Four of the six send real work to a real quantum computer. All six are free, and all six are open for anyone to copy and change.",[13,74153,74155],{"id":74154},"start-here","Start here",[18,74157,74158],{},"If you have never heard of Qollab or this challenge, you are exactly who this page is for. Two quick things to know before the tools.",[74160,74161],"fact-grid",{"b1":74162,"b2":74163,"t1":74164,"t2":74165},"A place to build quantum software in the open. You write a small quantum program in your browser, run it on a real quantum computer over the internet, and publish it so anyone can open it, copy it, and build on it. No lab and no hardware of your own required.","In spring 2026, Qollab teamed up with IonQ, a company that builds real quantum computers, and funded people around the world to make small open-source quantum projects, with time on IonQ's hardware. The results were grouped into four themes: music, art, finance, and this one.","What Qollab is","What the Creative Challenge was",[18,74167,74168,74169,74172],{},"The teams behind these six range from students to working quantum researchers. You will also see the phrase ",[154,74170,74171],{},"trapped-ion"," below. It is just one way of building a quantum computer, the kind IonQ makes, and the practical point is that four of these six tools run on the real machine and show you what comes back, noise and all. What ties all six together is a single instinct: rather than explain quantum computing, show it.",[13,74174,74176],{"id":74175},"what-are-these-six-tools","What are these six tools?",[74178,74179,74180,74183],"quick-answer",{},[18,74181,74182],{},"They are interactive quantum computing tools, things you operate rather than read. You step through an algorithm, build a circuit by hand, or run one on a real quantum computer and watch the result come back. All six came out of Qollab's Creative Challenge, all run in a browser, and all are open source.",[18,74184,74185],{},"QAVE animates an algorithm's state gate by gate. QuantumCanvas lets you drag and drop your own circuit. Quantum Courier is a game you play against a quantum solver. Quantum Advantage Lab races quantum against classical. qOrbital builds a molecule from a real quantum-chemistry run. QCFlows maps the correlations inside a circuit. Four run on real IonQ hardware. Pick whichever matches how you like to learn: watch, play, or build.",[13,74187,73838],{"id":73837},[41852,74189,74190,74206],{},[41855,74191,74192],{},[41858,74193,74194,74196,74199,74202,74204],{},[41861,74195,73850],{},[41861,74197,74198],{},"Type",[41861,74200,74201],{},"What you do with it",[41861,74203,73856],{},[41861,74205,73862],{},[41868,74207,74208,74224,74241,74257,74273,74289],{},[41858,74209,74210,74212,74215,74218,74221],{},[41873,74211,999],{},[41873,74213,74214],{},"Algorithm viewer",[41873,74216,74217],{},"Step through an algorithm and watch its state evolve",[41873,74219,74220],{},"Simulator",[41873,74222,74223],{},"Inho Choi",[41858,74225,74226,74229,74232,74235,74238],{},[41873,74227,74228],{},"QuantumCanvas",[41873,74230,74231],{},"Circuit sandbox",[41873,74233,74234],{},"Drag and drop operations into a working circuit",[41873,74236,74237],{},"IonQ hardware or simulator",[41873,74239,74240],{},"Shivani Mayekar",[41858,74242,74243,74246,74249,74252,74254],{},[41873,74244,74245],{},"Quantum Courier",[41873,74247,74248],{},"Game",[41873,74250,74251],{},"Beat, or lose to, a quantum solver at route planning",[41873,74253,74237],{},[41873,74255,74256],{},"Dr Siti Fariya",[41858,74258,74259,74262,74265,74268,74270],{},[41873,74260,74261],{},"Quantum Advantage Lab",[41873,74263,74264],{},"Quantum vs classical",[41873,74266,74267],{},"Race four algorithms against their classical rivals",[41873,74269,74237],{},[41873,74271,74272],{},"Hossein Sadeghi",[41858,74274,74275,74278,74281,74284,74286],{},[41873,74276,74277],{},"qOrbital",[41873,74279,74280],{},"Chemistry viewer",[41873,74282,74283],{},"Watch a molecule's orbital build up, noise and all",[41873,74285,74237],{},[41873,74287,74288],{},"Aryan Bawa & Arnav Singh",[41858,74290,74291,74294,74297,74300,74303],{},[41873,74292,74293],{},"QCFlows",[41873,74295,74296],{},"Correlation viewer",[41873,74298,74299],{},"See entanglement form and shift across a circuit",[41873,74301,74302],{},"Simulator (IonQ tomography planned)",[41873,74304,74305],{},"Paulo Itaboraí, Iosifina Angelidi & Kostas Blekos",[13,74307,74309],{"id":74308},"qave-watch-an-algorithm-run","QAVE: watch an algorithm run",[74022,74311],{"fork":74312,"showcase":1002,"who":74223},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fq-inho\u002Fqave",[18,74314,74315],{},"Start with the tool the whole category grew out of. QAVE, short for Quantum Algorithm Visualization Engine, was a winner of Qollab's very first challenge back in 2025. It takes a quantum algorithm and animates every step at once: the state, the underlying math, and the effect of each operation, all moving together as the algorithm runs. Most people are told what a quantum algorithm does. QAVE lets you watch it happen, and you can replay any circuit as many times as you like. It was built by Inho Choi, a quantum-information researcher, and it runs on a simulator, so nothing here needs special access.",[72443,74317,74319],{"avatar":104,"name":74223,"role":74318,"username":997},"Creator, QAVE",[18,74320,74321],{},"The most interesting thing for me has been seeing how much more understandable a quantum circuit and density matrix becomes once the evolution is made visible step by step. Many ideas in quantum computing feel difficult not only because of the math, but also because we often lack the right way to see them.",[13,74323,74325],{"id":74324},"quantumcanvas-build-one-by-hand","QuantumCanvas: build one by hand",[74022,74327],{"fork":74328,"showcase":74329,"who":74240},"https:\u002F\u002Fqollab.xyz\u002Fu\u002FShivaniMayekar\u002Fquantum-canvas","\u002Fexplore\u002Fquantum-canvas",[18,74331,74332],{},"QuantumCanvas is for the moment right after a tutorial, when you understand the idea but building a new circuit from scratch still feels like a wall. Instead of writing code, you drag and drop operations onto a grid, shake a qubit into superposition, link two together, and watch the circuit and its state react as you go. When you are ready, it runs the result on a real quantum computer, with no linear algebra required to begin. Shivani Mayekar, a researcher at Georgia Tech who has run quantum workshops for hundreds of people, built it around a single observation about how people click with the subject.",[72443,74334,74337],{"avatar":104,"name":74240,"role":74335,"username":74336},"Creator, QuantumCanvas","ShivaniMayekar",[18,74338,74339],{},"I love seeing the moment when something clicks. Someone who initially finds quantum computing intimidating suddenly becomes curious and engaged. Watching people discover that quantum is something they can explore rather than just admire from a distance is incredibly rewarding.",[13,74341,74343],{"id":74342},"quantum-courier-play-against-a-quantum-computer","Quantum Courier: play against a quantum computer",[74022,74345],{"fork":74346,"showcase":74347,"who":74256},"https:\u002F\u002Fqollab.xyz\u002Fu\u002FSitifar\u002Fquantum-game-pizza-race","\u002Fexplore\u002Fquantum-courier",[18,74349,74350],{},"Quantum Courier makes its case by letting you lose to a quantum computer. It is a browser game that turns delivery routing into a five-stage race: you draw your own routes, then a classical solver and a real quantum one both try to beat you, and the game shows honestly who wins each round and why. Quantum takes one stage on real IonQ hardware, while the classical solver still wins the routing rounds. Dr Siti Fariya, who spent two years optimising real traffic at the Port of Dover before moving into quantum, left every result in, wins and losses both, which is the honest thing to do and rarer than it should be.",[72443,74352,74355],{"avatar":104,"name":74256,"role":74353,"username":74354},"Creator, Quantum Courier","Sitifar",[18,74356,74357],{},"Quantum people are looking for real cases to solve, but they don't really understand the real problems in industry, like logistics. I want to be a bridge between the two.",[13,74359,74361],{"id":74360},"quantum-advantage-lab-watch-the-speedup-honestly","Quantum Advantage Lab: watch the speedup, honestly",[74022,74363],{"fork":74364,"showcase":74365,"who":74272},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fhsadeghi\u002Fquantum-advantage-lab","\u002Fexplore\u002Fquantum-advantage-lab",[18,74367,74368],{},"Everyone hears that quantum computers are faster. Far fewer ever see why. Quantum Advantage Lab runs four famous quantum algorithms side by side with the ordinary methods that solve the same problems, and streams the state of each one as it goes, so the reason a speedup exists becomes something you watch rather than a claim you accept. It is also refreshingly honest: at the small sizes today's hardware can handle, the quantum side does not actually win on a stopwatch, and the Lab says so out loud. Hossein Sadeghi, who spent a decade building quantum software at companies including D-Wave, built it on his own.",[72443,74370,74373],{"avatar":104,"name":74272,"role":74371,"username":74372},"Creator, Quantum Advantage Lab","hsadeghi",[18,74374,74375],{},"Quantum advantage is usually explained with asymptotic notation. That is technically correct, but it is not persuasive for most people. I built this to make the speedup something you can watch unfold, not just read about, even though in practice no such speedup exists yet.",[13,74377,74379],{"id":74378},"qorbital-chemistry-noise-and-all","qOrbital: chemistry, noise and all",[74022,74381],{"fork":74382,"showcase":74383,"who":74288},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fbawa27\u002Fqorbital","\u002Fexplore\u002Fqorbital",[18,74385,74386],{},"qOrbital is the most chemistry-flavoured of the six, and the most quietly radical. It runs a real quantum-chemistry calculation on IonQ hardware and draws the resulting molecule two different ways at once. Then it does something almost no visualiser does: instead of cleaning up the result into one tidy number, it overlays many real hardware runs so you can see exactly where the quantum computer is sure and where it is not. The noise is the exhibit, not something to hide. It was built by two Dartmouth physics students, Aryan Bawa and Arnav Singh.",[72443,74388,74391],{"avatar":104,"name":74288,"role":74389,"username":74390},"Creators, qOrbital","bawa27",[18,74392,74393],{},"Most VQE demos report one energy number and hide the noise. qOrbital makes the noise the exhibit.",[13,74395,74397],{"id":74396},"qcflows-see-the-correlations","QCFlows: see the correlations",[74022,74399],{"fork":74400,"showcase":74401,"who":74305},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fitaborala\u002Fqcflows","\u002Fexplore\u002Fqcflows",[18,74403,74404],{},"QCFlows is the one to reach for once the others have hooked you. A normal circuit diagram shows the gates but says nothing about the thing that makes a circuit quantum: the web of correlations between qubits that forms and shifts as it runs. QCFlows draws that web live, as a graph you can scrub through layer by layer, so something usually left to equations becomes visible. It comes from three researchers in the QUEST group at the Cyprus Institute, and it is deliberately built to be taken apart, so you can fork the whole dashboard or lift out just the one measurement you need.",[72443,74406,74410],{"avatar":104,"name":74407,"role":74408,"username":74409},"Paulo Itaboraí","Project lead, QCFlows","itaborala",[18,74411,74412],{},"We keep coming back to how difficult it actually is to visualize entanglement. The project is based on quantum tomography ideas, but those often stay on the academic side and don't get across to a general audience. We're trying to bring the visualization of correlations between qubits to a general audience.",[13,74414,74416],{"id":74415},"why-this-is-unusual","Why this is unusual",[18,74418,74419,74420,74424],{},"Learning quantum computing is not hard to find. ",[49,74421,74423],{"href":74422},"https:\u002F\u002Flearning.quantum.ibm.com","IBM",", Microsoft, PennyLane, Brilliant, and a shelf of university courses will all teach you the theory, and most are free. Almost all of it, though, is something you read or a course you sit through.",[18,74426,74427],{},"Tools you actually operate are rarer, and the good ones are scattered and often half-abandoned: a circuit editor here, a frozen game there, a simulator whose makers moved on. A curated set of them, built by named people, open to fork, and running on real quantum hardware, barely exists anywhere else.",[18,74429,74430,74431,74435],{},"That is what this page is. Six tools, six teams, one challenge. IonQ funds the hardware through its ",[49,74432,74434],{"href":74433},"https:\u002F\u002Fwww.ionq.com\u002Fnews\u002Fionq-partners-with-qollabs-creative-challenge-advancing-creative-quantum-innovation","partnership with Qollab",", and every project is open source, so the surest way to understand any of them is to open it and change something.",[13,74437,74094],{"id":74093},[18,74439,74440],{},"Every tool here runs in your browser, and every one is open to fork. Pick whichever one pulled you in, open it, and change something. That is usually the moment it clicks.",[74099,74442,74446],{"f1":74443,"l1":74444,"title":74445},"\u002Fexplore\u002F","Browse the full gallery","Open one and run it.",[18,74447,74448,74449],{},"Six tools, all open source, most on real IonQ hardware. Fork the one that caught your eye, or browse the full gallery. ",[154,74450,73040],{},{"title":104,"searchDepth":105,"depth":105,"links":74452},[74453,74454,74455,74456,74457,74458,74459,74460,74461,74462,74463],{"id":74154,"depth":105,"text":74155},{"id":74175,"depth":105,"text":74176},{"id":73837,"depth":105,"text":73838},{"id":74308,"depth":105,"text":74309},{"id":74324,"depth":105,"text":74325},{"id":74342,"depth":105,"text":74343},{"id":74360,"depth":105,"text":74361},{"id":74378,"depth":105,"text":74379},{"id":74396,"depth":105,"text":74397},{"id":74415,"depth":105,"text":74416},{"id":74093,"depth":105,"text":74094},[112,969,73752],[],{"username":8,"name":73075,"role":73076,"avatar":73077},"Quantum computing is famously hard to picture. Six people from Qollab's Creative Challenge each built a tool that fixes that a different way, something you open in a browser and watch, play, or take apart. Here is what they made, who made it, and how it works.","Six interactive quantum computing tools from Qollab's challenge, for newcomers: visualizers, a browser game, a circuit sandbox. Most run on real IonQ hardware.",{},"\u002Fblog\u002Feducation",[],{"title":74473,"description":74474},"Interactive Quantum Computing Tools","Six tools from Qollab's Creative Challenge that let you watch quantum computing happen and run it yourself. Written for people new to quantum.","blog\u002Feducation",[74477,143],"education","4dkg_eDLZZCCEir0iRjD-640gLFDlWCcVwH4m8_osgU",{"id":74480,"title":74481,"authors":74482,"body":74483,"breadcrumb":75051,"builders":75053,"byline":75054,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":75055,"description":75056,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":74128,"lessonCount":116,"meta":75057,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":75058,"publishDate":75059,"readingTime":75060,"related":75061,"relatedProjects":116,"seo":75062,"stem":75065,"tags":75066,"track":116,"trackName":116,"__hash__":75070},"blog\u002Fblog\u002Ffinance.md","What Can Quantum Computing Do in Finance?",[8],{"type":10,"value":74484,"toc":75041},[74485,74488,74491,74495,74498,74577,74581,74589,74607,74615,74681,74685,74688,74802,74805,74809,74822,74840,74853,74857,74865,74878,74891,74894,74898,74901,74904,74909,74918,74922,74926,74929,74932,74987,74995,74998,75002,75005,75012,75018,75027,75030],[18,74486,74487],{},"Quantum computing is further along in finance than in almost any other field, and further from paying off than the headlines suggest. The theory is a decade deep, the largest banks run dedicated research teams, and there is still no quantum computer that beats an ordinary one at a real financial task.",[18,74489,74490],{},"This page is a map of that gap. What quantum finance actually is, the handful of algorithms it rests on, what the banks have really shown, and an honest answer to whether it works yet. Then the part almost nobody offers: three open, forkable projects from Qollab's Spring 2026 challenge that let you run a piece of it yourself.",[13,74492,74494],{"id":74493},"the-three-projects","The three projects",[18,74496,74497],{},"Three open-source quantum finance projects came out of the Spring 2026 challenge, each taking on a different classic problem. Quantum Systemic Oracle runs a portfolio-optimization circuit on IonQ and publishes the result on-chain as a risk score. Quantum Regime Radar uses quantum kernels to match live markets against historical stress regimes. Quantum Market Game turns the prisoner's dilemma into a two-qubit game where entanglement changes the outcome. All three are forkable, run on real hardware, and are candid about where quantum helps and where it does not.",[41852,74499,74500,74514],{},[41855,74501,74502],{},[41858,74503,74504,74506,74509,74512],{},[41861,74505,73850],{},[41861,74507,74508],{},"What it does",[41861,74510,74511],{},"Under the hood",[41861,74513],{},[41868,74515,74516,74537,74557],{},[41858,74517,74518,74525,74528,74531],{},[41873,74519,74520,74524],{},[49,74521,74523],{"href":74522},"\u002Fexplore\u002Fquantum-systemic-oracle","Quantum Systemic Oracle"," · Jamie Dominguez",[41873,74526,74527],{},"A daily systemic-risk score from a portfolio-optimization circuit, published on-chain for smart contracts to read.",[41873,74529,74530],{},"14-qubit QAOA, Qiskit + IonQ, Chainlink oracle",[41873,74532,74533],{},[49,74534,74536],{"href":74535},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fjamie\u002Fquantum-systemic-oracle","Fork ⎆",[41858,74538,74539,74546,74549,74552],{},[41873,74540,74541,74545],{},[49,74542,74544],{"href":74543},"\u002Fexplore\u002Fquantum-regime-radar","Quantum Regime Radar"," · Alireza Khodaei",[41873,74547,74548],{},"Scores a live market against five volatility regimes taken from real history, by quantum-kernel overlap.",[41873,74550,74551],{},"12-qubit kernel, amplitude encoding, IonQ Forte",[41873,74553,74554],{},[49,74555,74536],{"href":74556},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Falireza\u002Fquantum-regime-radar",[41858,74558,74559,74566,74569,74572],{},[41873,74560,74561,74565],{},[49,74562,74564],{"href":74563},"\u002Fexplore\u002Fquantum-market-game","Quantum Market Game"," · Aadarsh Venkat Ramanan",[41873,74567,74568],{},"The prisoner's dilemma as two entangled qubits, reaching outcomes classical game theory cannot.",[41873,74570,74571],{},"2-qubit circuit, Qiskit + Streamlit, IonQ",[41873,74573,74574],{},[49,74575,74536],{"href":74576},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fq-aad\u002Fquantum-market-game-theory-sim",[13,74578,74580],{"id":74579},"where-quantum-finance-comes-from","Where quantum finance comes from",[18,74582,74583,74584,74588],{},"The idea is older than the hardware. In 1999, physicists Eisert, Wilkens, and Lewenstein showed that if you let players use ",[49,74585,74587],{"href":74586},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fquant-ph\u002F9806088","quantum strategies",", the prisoner's dilemma stops being a dilemma. The result comes with a caveat worth keeping: it holds only for a restricted set of strategies, and later work showed it does not survive once every quantum move is allowed. It was a hint, not a proof of advantage.",[18,74590,74591,74592,74596,74597,74601,74602,74606],{},"The finance-specific theory arrived in the 2010s. Ashley Montanaro proved a ",[49,74593,74595],{"href":74594},"https:\u002F\u002Farxiv.org\u002Fabs\u002F1504.06987","near-quadratic quantum speedup"," for Monte Carlo estimation, the workhorse behind pricing and risk. Rebentrost and colleagues turned that into the ",[49,74598,74600],{"href":74599},"https:\u002F\u002Farxiv.org\u002Fabs\u002F1805.00109","first quantum derivatives-pricing algorithm"," in 2018, and the 2019 ",[49,74603,74605],{"href":74604},"https:\u002F\u002Farxiv.org\u002Fabs\u002F1807.03890","Orus, Mugel, and Lizaso survey"," mapped the field: annealers for portfolios, arbitrage, and credit scoring, and amplitude estimation for pricing and risk.",[18,74608,74609,74610,74614],{},"Industry followed the theory. D-Wave, IBM, and IonQ put real machines within reach, and by 2020 a JPMorgan and IBM team had published a method to ",[49,74611,74613],{"href":74612},"https:\u002F\u002Fquantum-journal.org\u002Fpapers\u002Fq-2020-07-06-291\u002F","price options on a gate-based quantum computer",". That short arc, from a game-theory curiosity to a bank paper, is the whole prehistory.",[74616,74617,74619],"repo-spec",{"lead":74618},"A short timeline of the field.",[41852,74620,74621,74631],{},[41855,74622,74623],{},[41858,74624,74625,74628],{},[41861,74626,74627],{},"Field",[41861,74629,74630],{},"Detail",[41868,74632,74633,74641,74649,74657,74665,74673],{},[41858,74634,74635,74638],{},[41873,74636,74637],{},"1999",[41873,74639,74640],{},"Quantum game theory: entangled strategies change the prisoner's dilemma.",[41858,74642,74643,74646],{},[41873,74644,74645],{},"2015",[41873,74647,74648],{},"Montanaro proves a near-quadratic quantum speedup for Monte Carlo.",[41858,74650,74651,74654],{},[41873,74652,74653],{},"2018",[41873,74655,74656],{},"Rebentrost et al. give the first quantum algorithm for derivatives pricing.",[41858,74658,74659,74662],{},[41873,74660,74661],{},"2019",[41873,74663,74664],{},"The Orus survey maps quantum finance into its now-standard use cases.",[41858,74666,74667,74670],{},[41873,74668,74669],{},"2020",[41873,74671,74672],{},"JPMorgan and IBM publish option pricing on a gate-based quantum computer.",[41858,74674,74675,74678],{},[41873,74676,74677],{},"2021",[41873,74679,74680],{},"The Chakrabarti threshold paper estimates how far off real advantage is.",[13,74682,74684],{"id":74683},"what-quantum-computers-might-do-in-finance","What quantum computers might do in finance",[18,74686,74687],{},"The use cases are well mapped, and each rests on one of a few quantum algorithms. The honest-status column is the part most write-ups leave out. Read it as promise, not product: most of these are quadratic speedups that only pay off on fault-tolerant machines far larger than today's, and on the machine-learning side, classical methods often keep pace.",[41852,74689,74690,74703],{},[41855,74691,74692],{},[41858,74693,74694,74697,74700],{},[41861,74695,74696],{},"Use case",[41861,74698,74699],{},"Quantum approach",[41861,74701,74702],{},"Honest status",[41868,74704,74705,74719,74730,74741,74752,74766,74780,74791],{},[41858,74706,74707,74713,74716],{},[41873,74708,74709],{},[49,74710,74712],{"href":74711},"#a-risk-score-smart-contracts-can-read","Portfolio optimization",[41873,74714,74715],{},"QAOA, quantum annealing, VQE",[41873,74717,74718],{},"Small hardware demos; no edge over classical solvers yet.",[41858,74720,74721,74724,74727],{},[41873,74722,74723],{},"Derivatives pricing",[41873,74725,74726],{},"Quantum amplitude estimation",[41873,74728,74729],{},"A quadratic speedup in theory; needs fault-tolerant machines far beyond today's.",[41858,74731,74732,74735,74738],{},[41873,74733,74734],{},"Risk analysis (VaR, CVaR)",[41873,74736,74737],{},"Amplitude estimation",[41873,74739,74740],{},"Same quadratic speedup, same fault-tolerance requirement.",[41858,74742,74743,74746,74749],{},[41873,74744,74745],{},"Fraud and credit scoring",[41873,74747,74748],{},"Quantum kernels, QSVM",[41873,74750,74751],{},"Runs now at small scale; classical methods usually match it.",[41858,74753,74754,74760,74763],{},[41873,74755,74756],{},[49,74757,74759],{"href":74758},"#which-kind-of-market-is-this","Market-regime detection",[41873,74761,74762],{},"Quantum kernels",[41873,74764,74765],{},"Active research; honest projects show where it helps and where it does not.",[41858,74767,74768,74774,74777],{},[41873,74769,74770],{},[49,74771,74773],{"href":74772},"#game-theory-entangled","Game theory",[41873,74775,74776],{},"Entangled multi-qubit games",[41873,74778,74779],{},"A teaching lens more than a trading tool; entanglement shifts the equilibria.",[41858,74781,74782,74785,74788],{},[41873,74783,74784],{},"Synthetic market data",[41873,74786,74787],{},"QCBM, QGAN",[41873,74789,74790],{},"Research demos generate correlated returns for backtesting.",[41858,74792,74793,74796,74799],{},[41873,74794,74795],{},"Security",[41873,74797,74798],{},"Post-quantum cryptography",[41873,74800,74801],{},"The near-term reality: banks are migrating to quantum-safe encryption.",[18,74803,74804],{},"Two patterns run through the table. The pricing and risk methods, built on amplitude estimation and quantum Monte Carlo, are real and provable, but the speedup is quadratic rather than exponential and needs error-corrected hardware. The learning methods, quantum kernels and QSVM, run on today's machines, yet on ordinary financial data a well-tuned classical model usually matches them, a pattern researchers call dequantization. The one exception that already matters is security, and it is a threat rather than a speedup.",[13,74806,74808],{"id":74807},"who-is-actually-doing-it","Who is actually doing it",[18,74810,74811,74812,74816,74817,74821],{},"The clearest tell about quantum finance is that the bank doing the most is also the one publishing the doubts. JPMorgan's applied-research group, led by Marco Pistoia, co-wrote the 2020 option-pricing paper and keeps ",[49,74813,74815],{"href":74814},"https:\u002F\u002Fwww.cnbc.com\u002F2025\u002F07\u002F21\u002Fjpmorgan-quantum-computing-leadership-state-street-exec.html","investing in the area",". The same group also published ",[49,74818,74820],{"href":74819},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2211.16551","numerical evidence against"," quantum-kernel advantage on classical data. A team that argues against its own hype is a good signal.",[18,74823,74824,74825,74829,74830,74834,74835,74839],{},"Others are running real experiments at small scale. IBM and HSBC co-authored a 2023 study using ",[49,74826,74828],{"href":74827},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00260","quantum kernels for fraud and credit classification",", and in 2025 HSBC reported what it called the ",[49,74831,74833],{"href":74832},"https:\u002F\u002Fwww.hsbc.com\u002Fnews-and-views\u002Fnews\u002Fmedia-releases\u002F2025\u002Fhsbc-demonstrates-worlds-first-known-quantum-enabled-algorithmic-trading-with-ibm","first known quantum-enabled algorithmic bond trading"," trial. IonQ and Fidelity's applied-technology center generated ",[49,74836,74838],{"href":74837},"https:\u002F\u002Fwww.ionq.com\u002Fresources\u002Fgenerative-quantum-machine-learning-for-finance","synthetic market data"," on trapped-ion hardware. In every case the authors call the results early and scale-limited.",[18,74841,74842,74843,74847,74848,74852],{},"Not everyone is leaning in. More than fifteen banks have ",[49,74844,74846],{"href":74845},"https:\u002F\u002Fthequantuminsider.com\u002F2026\u002F03\u002F27\u002F15-plus-global-banks-probing-the-wonderful-world-of-quantum-technologies\u002F","active quantum programs",", but the commitment varies, and Goldman Sachs, a co-author of the field's key resource-estimate paper, has ",[49,74849,74851],{"href":74850},"https:\u002F\u002Fwww.bloomberg.com\u002Fnews\u002Ffeatures\u002F2026-04-26\u002Fwall-street-s-quantum-computing-divide-goldman-retreats-jpmorgan-invests","reportedly scaled back"," its quantum team. The common thread is that all of this lives inside corporate research. None of it ships to a developer.",[13,74854,74856],{"id":74855},"does-it-work-yet-an-honest-answer","Does it work yet? An honest answer",[18,74858,74859,74860,74864],{},"No, not in the sense that matters. There is no demonstrated production quantum advantage in finance today. The speedups that do exist are quadratic, not exponential, and they need fault-tolerant machines far beyond current noisy hardware. The most-cited estimate, a 2021 ",[49,74861,74863],{"href":74862},"https:\u002F\u002Fquantum-journal.org\u002Fpapers\u002Fq-2021-06-01-463\u002F","threshold paper"," from Goldman Sachs and IBM authors, put pricing a real derivative at roughly 8,000 logical qubits and a circuit depth in the tens of millions, and called it out of reach of current systems. A 2024 method trimmed that estimate, but not to anything you can run this decade.",[18,74866,74867,74868,74872,74873,74877],{},"The machine-learning side has its own reality check. When researchers, including JPMorgan's own, test quantum kernels on ordinary financial data, a well-tuned classical model tends to catch up, and a 2024 ",[49,74869,74871],{"href":74870},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.12618","review of quantum ML for finance"," concedes there is no provable exponential advantage for the near-term methods. Large value projections exist, such as BCG's estimate of up to ",[49,74874,74876],{"href":74875},"https:\u002F\u002Fwww.bcg.com\u002Fpress\u002F18july2024-quantum-computing-create-up-to-850-billion-of-economic-value-2040","$850 billion in economic value by 2040",", but that is a forecast across industries, not revenue anyone is earning now.",[18,74879,74880,74881,74885,74886,74890],{},"One area is already real, and it is a threat rather than a speedup. A future quantum computer could break the encryption that protects transactions, so data harvested today could be decrypted later. That is why banks are starting to adopt the ",[49,74882,74884],{"href":74883},"https:\u002F\u002Fwww.nist.gov\u002Fnews-events\u002Fnews\u002F2024\u002F08\u002Fnist-releases-first-3-finalized-post-quantum-encryption-standards","post-quantum encryption standards"," NIST finalized in 2024. One warning on names: a “quantum financial system” (QFS) and “Quantum AI” auto-trading platforms are ",[49,74887,74889],{"href":74888},"https:\u002F\u002Fpostquantum.com\u002Fquantum-snake-oil\u002Fquantum-financial-system\u002F","scams and conspiracy theories",", with no connection to any of the research above.",[18,74892,74893],{},"So where does that leave a developer who wants to touch this rather than read another projection? Not inside a bank lab. The three projects below are the accessible counterpoint, and together they map the whole on-ramp: a PhD whose dissertation is quantum finance, a bank data-governance veteran who had never run a quantum job, and a high schooler with a good mentor. None of them claims an advantage. Each is honest about its limits, which is exactly what makes them worth running.",[13,74895,74897],{"id":74896},"a-risk-score-smart-contracts-can-read","A risk score smart contracts can read",[18,74899,74900],{},"Jamie Dominguez spent more than a decade in enterprise data governance at a global bank. The Quantum Systemic Oracle was his first QPU job and his first Qiskit run. It takes a portfolio-optimization circuit, runs it on IonQ, distills the result into a single systemic-risk index in basis points, and publishes that number on-chain through a Chainlink-shaped interface, so any smart contract can read it the way it reads a price feed.",[18,74902,74903],{},"The framing, in his own words, is quantum compute as an on-chain primitive: not a dashboard people look at, but a number other code is built on. He is candid that AI pair-programming is what closed the gap between his finance background and the unfamiliar quantum and blockchain stacks, and he treats that as the point, a working loop you learn from by running it.",[831,74905],{"caption":74906,"no":835,"poster":74907,"video":74908},"One daily run end to end: live market data in, a QAOA job on IonQ, a systemic-risk index in basis points, and the on-chain publication. Press play.","\u002F_content\u002Fimages\u002Fsystemic-oracle\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002Fe2405685-d94e-4678-95ff-78c4a081aa9c",[72443,74910,74915],{"avatar":74911,"name":74912,"role":74913,"username":74914},"\u002F_content\u002Fimages\u002Fbuilders\u002Fjamie-dominguez.webp","Jamie Dominguez","Creator, Quantum Systemic Oracle","jamie",[18,74916,74917],{},"Given that quantum advancements are showing promise for financial applications like portfolio optimization, it was a perfect use case to blend my interests.",[74022,74919],{"fork":74535,"fork-label":74920,"showcase":74522,"showcase-label":74921},"Fork the oracle ⎆","Read the full story →",[13,74923,74925],{"id":74924},"which-kind-of-market-is-this","Which kind of market is this?",[18,74927,74928],{},"A volatility model fitted in a calm market breaks in a crisis, so Alireza Khodaei built a tool that answers the prior question: which kind of market is this? Quantum Regime Radar scores live equity returns against five volatility regimes taken from real history, episodes like the 2017 melt-up or the run-up to the SVB collapse. Each regime is encoded as a quantum state, and a quantum kernel measures how strongly today's market overlaps each one. His doctoral research put exactly this GARCH estimation on quantum hardware, so the reference library rests on real backtesting rather than hype.",[18,74930,74931],{},"What sets the project apart is the honesty. Before claiming anything for the quantum side, the team built the classical case against themselves, a plain correlation study that topped out too weak to use, and shipped it in the project so you can see the baseline for yourself.",[74616,74933,74935],{"lead":74934},"Five reference regimes, each anchored to a real market episode.",[41852,74936,74937,74945],{},[41855,74938,74939],{},[41858,74940,74941,74943],{},[41861,74942,74627],{},[41861,74944,74630],{},[41868,74946,74947,74955,74963,74971,74979],{},[41858,74948,74949,74952],{},[41873,74950,74951],{},"Complacency",[41873,74953,74954],{},"SPY 2017, the low-volatility melt-up.",[41858,74956,74957,74960],{},[41873,74958,74959],{},"Pre-crisis",[41873,74961,74962],{},"KRE 2023, the buildup to the SVB collapse.",[41858,74964,74965,74968],{},[41873,74966,74967],{},"Hyper-crisis",[41873,74969,74970],{},"SPY 2020, the COVID crash.",[41858,74972,74973,74976],{},[41873,74974,74975],{},"Leverage crisis",[41873,74977,74978],{},"SPY 2020, the long COVID grind that followed.",[41858,74980,74981,74984],{},[41873,74982,74983],{},"Recovery",[41873,74985,74986],{},"SPY late 2022, the climb back from that year's rout.",[72443,74988,74992],{"avatar":104,"name":74989,"role":74990,"username":74991},"Alireza Khodaei","Creator, Quantum Regime Radar","alireza",[18,74993,74994],{},"With any quantum algorithm, especially in finance, the first question you get is: everything is fine with classical, why even bother with quantum? We wanted to show, in one picture, that classical is not delivering a tangible advantage here.",[74022,74996],{"fork":74556,"fork-label":74997,"showcase":74543,"showcase-label":74921},"Fork Regime Radar ⎆",[13,74999,75001],{"id":75000},"game-theory-entangled","Game theory, entangled",[18,75003,75004],{},"Aadarsh Venkat Ramanan, a rising high-school senior, found his way into quantum through Marco Pistoia, who heads JPMorgan's quantum research lab and had worked with his mother at the bank. His Quantum Market Game takes the classic prisoner's dilemma and runs it on a quantum computer: two traders are two qubits, each in a superposition of buy and sell, and an optional entangling gate links their choices. Turn entanglement on and the game settles into outcomes classical game theory cannot reach.",[18,75006,75007,75008,75011],{},"That idea has a serious lineage. The ",[49,75009,75010],{"href":74586},"Eisert-Wilkens-Lewenstein scheme"," showed in 1999 that a quantum prisoner's dilemma opens up equilibria the classical game never had. Aadarsh keeps his version deliberately small, a two-qubit circuit you can read in one sitting, because it is built as a teaching object: superposition is the trader who has not decided, entanglement is the toggle that ties two fates together.",[831,75013],{"caption":75014,"no":844,"poster":75015,"video":75016,"alt":75017},"The Quantum Market Game: set each trader's buy\u002Fsell odds, toggle entanglement, then run the market and read the payoffs.","\u002F_content\u002Fimages\u002Fquantum-market-game\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F38e61a99-e51d-4be0-86bf-afe6af5bdd9a","The Quantum Market Game simulator: two traders, Bull and Bear, each choosing a quantum strategy angle before the entangled game runs",[72443,75019,75024],{"avatar":75020,"name":75021,"role":75022,"username":75023},"\u002F_content\u002Fimages\u002Fbuilders\u002Faadarsh-venkat-ramanan.webp","Aadarsh Venkat Ramanan","Creator, Quantum Market Game","q-aad",[18,75025,75026],{},"People should care because this is a baseline for future quantum use cases in finance, and it lets people learn the basic principles of quantum mechanics in an intuitive way, creating curiosity and further learning.",[74022,75028],{"fork":74576,"fork-label":75029,"showcase":74563,"showcase-label":74921},"Fork the game ⎆",[74099,75031,75035],{"f1":75032,"l1":75033,"title":75034},"https:\u002F\u002Fqollab.xyz\u002Fnew","Start a project","Pick a problem in finance. Run it on real hardware.",[18,75036,75037,75038],{},"Every project here is open, forkable, and yours to build on. Start from one of these, or bring your own idea to the next challenge. ",[154,75039,75040],{},"Everything runs on real quantum hardware through Qollab.",{"title":104,"searchDepth":105,"depth":105,"links":75042},[75043,75044,75045,75046,75047,75048,75049,75050],{"id":74493,"depth":105,"text":74494},{"id":74579,"depth":105,"text":74580},{"id":74683,"depth":105,"text":74684},{"id":74807,"depth":105,"text":74808},{"id":74855,"depth":105,"text":74856},{"id":74896,"depth":105,"text":74897},{"id":74924,"depth":105,"text":74925},{"id":75000,"depth":105,"text":75001},[112,969,75052],"Finance",[],{"username":8,"name":73075,"role":73076,"avatar":73077},"Banks have chased quantum finance for a decade, and it still cannot beat an ordinary computer at a real money problem. Here is the honest state of the field, from portfolio risk to market games, and three Spring 2026 projects that let you run a piece of it yourself.","Quantum computing in finance, explained honestly: the history, the algorithms, what JPMorgan and HSBC have shown, and three open projects you can run yourself.",{},"\u002Fblog\u002Ffinance","2026-07-03","9 min read",[],{"title":75063,"description":75064},"Quantum Computing in Finance: A Practical, Honest Guide","The history, the real use cases and their algorithms, what banks like JPMorgan and HSBC have actually shown, and an honest read on quantum advantage.","blog\u002Ffinance",[75067,143,75068,75069],"finance","optimization","machine-learning","Ow3UwNb0zZhOEa_DBFBGud_XVBSj_PI4CgoQzJSC1JU",{"id":75072,"title":75073,"authors":75074,"body":75075,"breadcrumb":75387,"builders":75388,"byline":75389,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":75390,"description":75391,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":74128,"lessonCount":116,"meta":75392,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":75393,"publishDate":75059,"readingTime":74132,"related":75394,"relatedProjects":116,"seo":75395,"stem":75397,"tags":75398,"track":116,"trackName":116,"__hash__":75399},"blog\u002Fblog\u002Fmusic.md","What Does a Quantum Computer Sound Like?",[8],{"type":10,"value":75076,"toc":75376},[75077,75080,75083,75087,75090,75093,75095,75098,75163,75167,75170,75173,75176,75179,75186,75189,75193,75195,75203,75206,75219,75225,75228,75231,75238,75241,75245,75247,75250,75255,75258,75262,75265,75269,75272,75275,75280,75285,75288,75293,75297,75300,75307,75312,75317,75320,75323,75329,75333,75336,75343,75347,75350,75353,75357,75359,75362],[18,75078,75079],{},"A quantum computer's raw output is a probability distribution: a spread of numbers with no obvious shape. Turning that spread into sound is one of the more direct ways to make quantum behavior something you can actually perceive, not just calculate.",[18,75081,75082],{},"It is also not a new idea. Quantum computer music has been a real field for years, mostly out of reach unless you had lab access or a physics PhD. Three teams from Qollab's Spring 2026 Creative Challenge changed that this year, and built three different answers to what it sounds like.",[13,75084,75086],{"id":75085},"what-is-quantum-computing-music","What is quantum computing music?",[18,75088,75089],{},"Quantum computing music turns a real quantum circuit's output into sound: its probabilities, amplitudes, phase, and measurement outcomes become the raw material. That is different from using a quantum computer as a fancy random-number generator.",[18,75091,75092],{},"On Qollab, three Spring 2026 teams built three different answers to what that sounds like. Musiq maps circuit data directly to sound as a teaching instrument. Quantum Patterns turns quantum cellular automata into live-coded compositional material. Superposition Sequencer plays user-designed circuits like a synthesizer. All three run on real IonQ hardware, not a simulator dressed up, and all three are open source. Start with whichever metaphor sounds most like you: translator, material, or instrument.",[13,75094,73838],{"id":73837},[18,75096,75097],{},"Three teams, three instruments, one starting question. Here is who built what.",[41852,75099,75100,75116],{},[41855,75101,75102],{},[41858,75103,75104,75106,75109,75111,75114],{},[41861,75105,73850],{},[41861,75107,75108],{},"The metaphor",[41861,75110,73856],{},[41861,75112,75113],{},"What you hear",[41861,75115,73862],{},[41868,75117,75118,75133,75149],{},[41858,75119,75120,75122,75125,75127,75130],{},[41873,75121,73103],{},[41873,75123,75124],{},"Translator",[41873,75126,74237],{},[41873,75128,75129],{},"Circuit data mapped straight to frequency, loudness, and texture",[41873,75131,75132],{},"Tomoya Hatanaka & Emmanuella Adams",[41858,75134,75135,75137,75140,75143,75146],{},[41873,75136,73744],{},[41873,75138,75139],{},"Material",[41873,75141,75142],{},"IonQ hardware or local statevector",[41873,75144,75145],{},"Quantum cellular automata, live-coded into pitch, rhythm, and space",[41873,75147,75148],{},"Peter Thomas & Paulo Itaboraí",[41858,75150,75151,75153,75156,75158,75161],{},[41873,75152,73096],{},[41873,75154,75155],{},"Instrument",[41873,75157,74237],{},[41873,75159,75160],{},"A step sequencer where each shot of your circuit is a beat",[41873,75162,73132],{},[13,75164,75166],{"id":75165},"why-sound","Why sound?",[18,75168,75169],{},"Quantum mechanics is usually taught through equations: wavefunctions, probability amplitudes, measurement operators. That is precise, but it asks a lot of anyone without a physics background, and even physicists often reach for a second way to build intuition. Sound is one option.",[18,75171,75172],{},"People are good at hearing structure. A chord, a shift in rhythm, a change in timbre register instantly, without translation. When a circuit's measurement outcomes become audible, properties that are hard to picture on paper turn into things you notice by ear.",[18,75174,75175],{},"A spread of possible outcomes can sound like a chord collapsing into one note. Two qubits that stay correlated can sound like two voices moving together for no obvious reason. Interference can sound like loudness rising and falling as amplitudes reinforce or cancel.",[18,75177,75178],{},"Tomoya Hatanaka, who built Musiq to turn circuit data straight into sound, designed the whole project around that idea.",[72443,75180,75183],{"avatar":104,"name":75181,"role":75182,"username":73101},"Tomoya Hatanaka","Project lead, Musiq",[18,75184,75185],{},"Music serves as a universal translator, allowing users to intuitively hear complex quantum concepts like superposition and entanglement without relying on mathematical formulas.",[18,75187,75188],{},"Francisco Estivallet, who built Superposition Sequencer and came to quantum through creative coding rather than physics, describes the same effect from the listener's side.",[72443,75190,75191],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,75192,73668],{},[13,75194,73949],{"id":73948},[18,75196,75197,75198,75202],{},"For most of quantum computer music's history, taking part meant institutional access. A composer and researcher named Eduardo Reck Miranda founded the field at Plymouth's ",[49,75199,75201],{"href":75200},"https:\u002F\u002Fwww.plymouth.ac.uk\u002Fresearch\u002Ficcmr","Interdisciplinary Centre for Computer Music Research"," in the early 2020s, working from inside a university lab with its own dedicated quantum hardware.",[18,75204,75205],{},"He released an album composed with a quantum computer and built a toolkit for musicians. Peter Thomas, who later built Quantum Patterns, trained in that same lab.",[18,75207,75208,75209,75213,75214,75218],{},"The field grew into its own conference, the ",[49,75210,75212],{"href":75211},"https:\u002F\u002F2025.isqcmc.org\u002F","International Symposium on Quantum Computing and Musical Creativity",", which moved from Plymouth in 2021 to Berlin in 2023 to Palermo in 2025, the same year UNESCO named its International Year of Quantum Science and Technology. A growing ",[49,75215,75217],{"href":75216},"https:\u002F\u002Fpubmed.ncbi.nlm.nih.gov\u002F38996413\u002F","academic literature"," followed, alongside coverage from outlets like Physics World and IBM's own research blog. Almost none of it was something you could open in a browser and try.",[18,75220,75221,75222,75224],{},"What changed is access to the hardware itself. Cloud quantum computers you can run a real circuit on from a browser tab are a recent development. ",[154,75223,112],{}," is a community and coding platform built around that shift: a place to write and run quantum code with direct access to IonQ's trapped-ion machines, no lab or university affiliation required.",[18,75226,75227],{},"In spring 2026, Qollab and IonQ funded a Creative Challenge on top of that platform: compute credits, cash, and mentorship for open, original projects from anyone with an idea. Musiq, Quantum Patterns, and Superposition Sequencer came out of that program, alongside a dozen other projects spanning art, finance, and education. All three are open source and forkable, and built not only by physicists: Francisco is a mechatronics engineer, Emmanuella a software-engineering student.",[18,75229,75230],{},"Peter puts what that access changes in personal terms.",[72443,75232,75235],{"avatar":104,"name":75233,"role":75234,"username":73742},"Peter Thomas","Project lead, Quantum Patterns",[18,75236,75237],{},"It can disseminate knowledge in a way that a paper can't.",[18,75239,75240],{},"Francisco puts the case for engaging with it now more simply.",[72443,75242,75243],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,75244,73694],{},[13,75246,73998],{"id":73997},[18,75248,75249],{},"What is built so far is a first pass. Each project currently maps one circuit's output to one piece of sound or one round of a pattern. The ambition across all three is to scale that up as the hardware and the techniques mature.",[72443,75251,75252],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,75253,75254],{},"We will scale up to 30 to 40 qubits on IonQ hardware, expanding from simple sound generation to the automated composition of fully structured music.",[18,75256,75257],{},"Others are thinking about the interface itself. Francisco wants a version of Superposition Sequencer you could play like a physical instrument, not just a browser tab, and is still refining how the hardware's own imperfections should sound.",[72443,75259,75260],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,75261,73686],{},[18,75263,75264],{},"The larger bet is the same one live coding and early electronic music made: that a new way to make sound pulls in people who would never open a physics textbook, and some of them stay long enough to understand the hardware underneath. If quantum computer music follows that path, its biggest effect may not be the music at all. It may be who ends up learning to think in qubits because a synthesizer got them curious first.",[13,75266,75268],{"id":75267},"musiq-quantum-data-as-sound","Musiq: quantum data as sound",[74022,75270],{"fork":75271,"showcase":73105,"who":75132},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fdoraking\u002Fmusiq",[18,75273,75274],{},"Musiq is a browser-based sonification studio built by Tomoya Hatanaka, a freelance quantum engineer, and Emmanuella Adams, a creative technologist. You build a circuit, run it on a simulator or real IonQ hardware, and Musiq maps the result directly onto sound: basis-state index becomes frequency, measurement probability becomes strength, amplitude becomes loudness, and phase becomes interference.",[72443,75276,75277],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,75278,75279],{},"I wanted to overcome the repetitive nature of classical algorithmic music by directly translating the mathematical spread of quantum states into dynamic musical expression.",[831,75281],{"alt":75282,"caption":104,"no":104,"poster":75283,"video":75284},"Musiq, a browser studio that turns quantum circuits into sound","\u002F_content\u002Fimages\u002Fmusiq\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F064dd3e0-1e11-4ebc-8646-a5b4444fbbca",[18,75286,75287],{},"The mapping is literal enough that different circuits sound genuinely different, and for Tomoya the sound is also a small argument about what quantum computers are for. Most of the field points its hardware at optimization and simulation.",[72443,75289,75290],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,75291,75292],{},"It proves that quantum computing can expand beyond pragmatic optimization or calculation tasks, capturing new potential for creative and artistic expression.",[13,75294,75296],{"id":75295},"quantum-patterns-circuits-as-material","Quantum Patterns: circuits as material",[74022,75298],{"fork":75299,"showcase":73746,"who":75148},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fcephasteom\u002Fquantum-patterns",[18,75301,75302,75303,75306],{},"Quantum Patterns comes from Peter Thomas, who performs as ",[1031,75304,75305],{},"Cephas Teom",", with Paulo Itaboraí on the quantum side. Partitioned quantum cellular automata run on a circuit, and their measurement outcomes become raw material for a fork of Satori, Peter's browser-based live-coding environment, where short scripts turn the data into pitch, rhythm, timbre, and space.",[72443,75308,75309],{"avatar":104,"name":75233,"role":75234,"username":73742},[18,75310,75311],{},"I was looking at ways to make quantum computer music, and more broadly the use of quantum in the arts, more widely adopted. I used the live coding scene as a blueprint: a culture that started in academia but now has broad appeal, because of its mature ecosystem of tools and a really supportive ethos.",[831,75313],{"alt":75314,"caption":104,"no":104,"poster":75315,"video":75316},"The Satori PQCA live-coding environment, a quantum cellular automaton visualized alongside its musical script","\u002F_content\u002Fimages\u002Fquantum-patterns\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002Fef35c47f-a7cc-4dae-9613-64db0498c750",[18,75318,75319],{},"Peter's doctorate at Plymouth's ICCMR, the lab that founded quantum computer music as an academic field, sits behind the project. He frames the results less as fixed songs than as systems of relationships and probabilities.",[18,75321,75322],{},"Paulo, who has spent years putting quantum algorithms on stage, sees the same opening for a wider audience.",[72443,75324,75326],{"avatar":104,"name":74407,"role":75325,"username":74409},"Quantum algorithms & hardware, Quantum Patterns",[18,75327,75328],{},"This platform is already really stage-tested, and I think it can reach a lot of people.",[13,75330,75332],{"id":75331},"superposition-sequencer-circuit-as-instrument","Superposition Sequencer: circuit as instrument",[74022,75334],{"fork":75335,"showcase":73099,"who":73132},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fincomputable\u002Fsuperposition-sequencer",[18,75337,75338,75339,75342],{},"Francisco Estivallet, a mechatronics engineer and creative technologist working as ",[1031,75340,75341],{},"Incomputable"," in Barcelona, built a sequencer where the notes are not programmed directly. A visual editor lets you design a circuit, and running it on IonQ hardware or a simulator drives pitch, rhythm, velocity, and timbre.",[72443,75344,75345],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,75346,73151],{},[831,75348],{"alt":75349,"caption":104,"no":104,"poster":73187,"video":73188},"Superposition Sequencer, a browser-based quantum music sequencer",[18,75351,75352],{},"Francisco's way in was history, not physics. He points to how early computing and electronics sparked a wave of musical exploration, and thinks quantum computing is due for a similar moment. Rather than fighting the hardware's imperfections, the project leans into them.",[72443,75354,75355],{"avatar":73131,"name":73132,"role":73133,"username":73094},[18,75356,73159],{},[13,75358,74094],{"id":74093},[18,75360,75361],{},"Every project here is open source, and you can run one in your browser right now. Start from whichever metaphor pulled you in: hear your own circuit as sound, live-code with quantum-generated patterns, or play a sequencer where the notes come from a quantum measurement.",[74099,75363,75371],{"c1":75364,"c2":75365,"c3":75366,"f1":75271,"f2":75299,"f3":75335,"l1":75367,"l2":75368,"l3":75369,"title":75370},"#46e0ff","#7fbcff","#9b7bff","Fork Musiq","Fork Quantum Patterns","Fork the Sequencer","Pick your instrument.",[18,75372,75373,75374],{},"Fork any of the three, run a circuit on real IonQ hardware, and hear what it does. ",[154,75375,73040],{},{"title":104,"searchDepth":105,"depth":105,"links":75377},[75378,75379,75380,75381,75382,75383,75384,75385,75386],{"id":75085,"depth":105,"text":75086},{"id":73837,"depth":105,"text":73838},{"id":75165,"depth":105,"text":75166},{"id":73948,"depth":105,"text":73949},{"id":73997,"depth":105,"text":73998},{"id":75267,"depth":105,"text":75268},{"id":75295,"depth":105,"text":75296},{"id":75331,"depth":105,"text":75332},{"id":74093,"depth":105,"text":74094},[112,969,73097],[],{"username":8,"name":73075,"role":73076,"avatar":73077},"Three Spring 2026 teams turned real IonQ quantum hardware into music: a translator, a set of live-coded patterns, and a playable instrument. This is quantum computing music, in the words of the people building it.","Three Spring 2026 teams turned real IonQ quantum hardware into music: a translator, a live-coding tool, and a playable sequencer. All open source.",{},"\u002Fblog\u002Fmusic",[],{"title":75396,"description":75391},"Quantum Computing Music","blog\u002Fmusic",[73758,143],"v8jsa6nn4hl9KcYm3DlIPi9PEuu2KQRgqkkbV0UhnFg",{"id":75401,"title":75402,"authors":75403,"body":75406,"breadcrumb":76211,"builders":76212,"byline":76236,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":76237,"description":76238,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":76239,"hero":76241,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":76244,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":76245,"publishDate":76246,"readingTime":993,"related":76247,"relatedProjects":76248,"seo":76256,"stem":76259,"tags":76260,"track":116,"trackName":116,"__hash__":76261},"blog\u002Fblog\u002Fqcflows.md","Project Showcase: QCFlows",[74409,75404,75405],"iosifinaangelidi","eelvex",{"type":10,"value":75407,"toc":76204},[75408,75411,75419,75422,75426,75430,75433,75480,75483,75490,75494,75502,76087,76098,76150,76154,76157,76160,76167,76171,76174,76179,76182,76187,76189,76192,76201],[18,75409,75410],{},"A circuit diagram shows you the gates. It says nothing about the thing that makes the circuit quantum: the web of correlations that forms, shifts, and spreads between qubits as the state evolves.",[18,75412,75413,75414,75418],{},"QCFlows makes that web visible. Draw a circuit onto an interactive qubit graph, or import your QASM, and a live dashboard renders the correlation structure as a dynamic network you can scrub through, layer by layer. It runs in the browser at ",[49,75415,75417],{"href":75416},"https:\u002F\u002Fapp.qcflows.net\u002F","app.qcflows.net",", and the whole stack is open source.",[18,75420,75421],{},"It is a Spring 2026 challenge project from three researchers in the QUEST group at the Cyprus Institute: Paulo Itaboraí, who leads the project, with Dr. Iosifina Angelidi on theory and Dr. Kostas Blekos on quantum information. The starting point is a frustration the group kept meeting in its own research.",[72443,75423,75424],{"avatar":104,"name":74407,"role":74408,"username":74409},[18,75425,74412],{},[13,75427,75429],{"id":75428},"seeing-past-the-gate-diagram","Seeing past the gate diagram",[18,75431,75432],{},"Most circuit tools stop at a static, gate-by-gate layout. The correlation structure that builds up while those gates run is invisible in that view, and it is precisely the part that carries the quantum behaviour. QCFlows puts it on screen: an interactive qubit graph and a metric-matrix heatmap sit alongside a traditional wire view, all linked to the live statevector, so every gate you add or remove redraws the whole picture.",[74616,75434,75436],{"lead":75435},"One dashboard, four linked views of the same state.",[41852,75437,75438,75446],{},[41855,75439,75440],{},[41858,75441,75442,75444],{},[41861,75443,74627],{},[41861,75445,74630],{},[41868,75447,75448,75456,75464,75472],{},[41858,75449,75450,75453],{},[41873,75451,75452],{},"Qubit graph",[41873,75454,75455],{},"The circuit drawn as a network; edges weight live pairwise correlations.",[41858,75457,75458,75461],{},[41873,75459,75460],{},"Metric matrix",[41873,75462,75463],{},"A heatmap of every pair under the chosen metric and basis.",[41858,75465,75466,75469],{},[41873,75467,75468],{},"Circuit timeline",[41873,75470,75471],{},"Scrub through the gate sequence; every view follows in real time.",[41858,75473,75474,75477],{},[41873,75475,75476],{},"Statevector readout",[41873,75478,75479],{},"The raw amplitudes behind the pictures.",[18,75481,75482],{},"The idea predates the challenge. As Kostas tells it, it came from all three of them at once, out of calculations and preliminary graphs they were already making about how information moves through a circuit.",[72443,75484,75487],{"avatar":104,"name":75485,"role":75486,"username":75405},"Dr. Kostas Blekos","Quantum information, QCFlows",[18,75488,75489],{},"The point, when we started this, was to get insight into how quantum algorithms work in the dynamic sense.",[13,75491,75493],{"id":75492},"a-direction-for-correlation","A direction for correlation",[18,75495,75496,75497,75501],{},"The dashboard's default lens is the team's own metric: the K-network, formalized in their June 2026 paper, ",[49,75498,75500],{"href":75499},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2606.16549","“What does measuring one qubit reveal about another?”"," It answers a question a symmetric weight cannot: if you measure qubit i in a given basis, how strongly does that outcome reshape the state of qubit j? The scoring engine behind it is compact enough to read in one sitting.",[493,75503,75506],{"name":75504,"run-href":75505,"tag":72511},"k_measure.py","https:\u002F\u002Fgithub.com\u002FItaborala\u002FQCFlows",[498,75507,75509],{"className":500,"code":75508,"language":502,"meta":72515,"style":104},"# How the backend scores a directed correlation, K[i->j], from a pair's\n# reduced density matrix. Excerpt from qcflows_api\u002Fk_measure.py.\nimport numpy as np\nfrom qiskit import QuantumCircuit\n\n# A 4-qubit ladder: each CX hands correlation one qubit down the line.\nCIRCUIT = QuantumCircuit(4)\nCIRCUIT.ry(1.2, 0)\nCIRCUIT.cx(0, 1)\nCIRCUIT.ry(1.2, 1)\nCIRCUIT.cx(1, 2)\nCIRCUIT.ry(1.2, 2)\nCIRCUIT.cx(2, 3)\n\ndef directional_k(rho2, zero_idx, one_idx, tol=1e-10):\n    \"\"\"K for measuring one qubit (Z basis) and inspecting the other.\"\"\"\n    s0 = rho2[np.ix_(zero_idx, zero_idx)]   # measured qubit read 0\n    s1 = rho2[np.ix_(one_idx, one_idx)]     # measured qubit read 1\n    p0, p1 = np.real(np.trace(s0)), np.real(np.trace(s1))\n    if p0 \u003C tol or p1 \u003C tol:\n        return 0.0\n    value = 4.0 * p0 * p1 * (1.0 - squared_fidelity(s0 \u002F p0, s1 \u002F p1))\n    return float(np.clip(value, 0.0, 1.0))\n\ndef directed_pair_k(rho2):\n    \"\"\"(K[i->j], K[j->i]) from one 2-qubit reduced density matrix.\"\"\"\n    k_ij = directional_k(rho2, [0, 1], [2, 3])   # measure i, inspect j\n    k_ji = directional_k(rho2, [0, 2], [1, 3])   # measure j, inspect i\n    return k_ij, k_ji\n\n# The flow: truncate the circuit after every gate and rescore every pair.\nfor m in range(len(CIRCUIT.data) + 1):\n    state = prefix_state(CIRCUIT, m)   # statevector after the first m gates\n    print(f\"marker {m}:\\n{k_matrix(state).round(2)}\")\n",[504,75510,75511,75516,75521,75531,75541,75545,75550,75565,75583,75601,75619,75637,75655,75673,75677,75711,75716,75735,75752,75780,75802,75808,75858,75879,75883,75896,75901,75942,75972,75979,75983,75988,76015,76043],{"__ignoreMap":104},[507,75512,75513],{"class":509,"line":510},[507,75514,75515],{"class":562},"# How the backend scores a directed correlation, K[i->j], from a pair's\n",[507,75517,75518],{"class":509,"line":105},[507,75519,75520],{"class":562},"# reduced density matrix. Excerpt from qcflows_api\u002Fk_measure.py.\n",[507,75522,75523,75525,75527,75529],{"class":509,"line":540},[507,75524,514],{"class":513},[507,75526,518],{"class":517},[507,75528,521],{"class":513},[507,75530,524],{"class":517},[507,75532,75533,75535,75537,75539],{"class":509,"line":553},[507,75534,529],{"class":513},[507,75536,532],{"class":517},[507,75538,514],{"class":513},[507,75540,537],{"class":517},[507,75542,75543],{"class":509,"line":559},[507,75544,556],{"emptyLinePlaceholder":133},[507,75546,75547],{"class":509,"line":566},[507,75548,75549],{"class":562},"# A 4-qubit ladder: each CX hands correlation one qubit down the line.\n",[507,75551,75552,75555,75557,75559,75561,75563],{"class":509,"line":590},[507,75553,75554],{"class":583},"CIRCUIT",[507,75556,1423],{"class":572},[507,75558,577],{"class":576},[507,75560,580],{"class":517},[507,75562,12152],{"class":583},[507,75564,587],{"class":517},[507,75566,75567,75569,75571,75573,75575,75577,75579,75581],{"class":509,"line":610},[507,75568,75554],{"class":583},[507,75570,53],{"class":517},[507,75572,639],{"class":576},[507,75574,580],{"class":517},[507,75576,23656],{"class":583},[507,75578,622],{"class":517},[507,75580,601],{"class":583},[507,75582,587],{"class":517},[507,75584,75585,75587,75589,75591,75593,75595,75597,75599],{"class":509,"line":634},[507,75586,75554],{"class":583},[507,75588,53],{"class":517},[507,75590,615],{"class":576},[507,75592,580],{"class":517},[507,75594,601],{"class":583},[507,75596,622],{"class":517},[507,75598,625],{"class":583},[507,75600,587],{"class":517},[507,75602,75603,75605,75607,75609,75611,75613,75615,75617],{"class":509,"line":661},[507,75604,75554],{"class":583},[507,75606,53],{"class":517},[507,75608,639],{"class":576},[507,75610,580],{"class":517},[507,75612,23656],{"class":583},[507,75614,622],{"class":517},[507,75616,625],{"class":583},[507,75618,587],{"class":517},[507,75620,75621,75623,75625,75627,75629,75631,75633,75635],{"class":509,"line":678},[507,75622,75554],{"class":583},[507,75624,53],{"class":517},[507,75626,615],{"class":576},[507,75628,580],{"class":517},[507,75630,625],{"class":583},[507,75632,622],{"class":517},[507,75634,584],{"class":583},[507,75636,587],{"class":517},[507,75638,75639,75641,75643,75645,75647,75649,75651,75653],{"class":509,"line":683},[507,75640,75554],{"class":583},[507,75642,53],{"class":517},[507,75644,639],{"class":576},[507,75646,580],{"class":517},[507,75648,23656],{"class":583},[507,75650,622],{"class":517},[507,75652,584],{"class":583},[507,75654,587],{"class":517},[507,75656,75657,75659,75661,75663,75665,75667,75669,75671],{"class":509,"line":697},[507,75658,75554],{"class":583},[507,75660,53],{"class":517},[507,75662,615],{"class":576},[507,75664,580],{"class":517},[507,75666,584],{"class":583},[507,75668,622],{"class":517},[507,75670,8226],{"class":583},[507,75672,587],{"class":517},[507,75674,75675],{"class":509,"line":710},[507,75676,556],{"emptyLinePlaceholder":133},[507,75678,75679,75681,75684,75686,75689,75691,75694,75696,75699,75701,75704,75706,75709],{"class":509,"line":715},[507,75680,1370],{"class":513},[507,75682,75683],{"class":576}," directional_k",[507,75685,580],{"class":517},[507,75687,75688],{"class":1382},"rho2",[507,75690,622],{"class":517},[507,75692,75693],{"class":1382},"zero_idx",[507,75695,622],{"class":517},[507,75697,75698],{"class":1382},"one_idx",[507,75700,622],{"class":517},[507,75702,75703],{"class":1382},"tol",[507,75705,573],{"class":517},[507,75707,75708],{"class":583},"1e-10",[507,75710,1883],{"class":517},[507,75712,75713],{"class":509,"line":721},[507,75714,75715],{"class":730},"    \"\"\"K for measuring one qubit (Z basis) and inspecting the other.\"\"\"\n",[507,75717,75718,75721,75723,75726,75729,75732],{"class":509,"line":736},[507,75719,75720],{"class":517},"    s0 ",[507,75722,573],{"class":572},[507,75724,75725],{"class":517}," rho2[np.",[507,75727,75728],{"class":576},"ix_",[507,75730,75731],{"class":517},"(zero_idx, zero_idx)]   ",[507,75733,75734],{"class":562},"# measured qubit read 0\n",[507,75736,75737,75740,75742,75744,75746,75749],{"class":509,"line":748},[507,75738,75739],{"class":517},"    s1 ",[507,75741,573],{"class":572},[507,75743,75725],{"class":517},[507,75745,75728],{"class":576},[507,75747,75748],{"class":517},"(one_idx, one_idx)]     ",[507,75750,75751],{"class":562},"# measured qubit read 1\n",[507,75753,75754,75757,75759,75761,75763,75765,75768,75771,75773,75775,75777],{"class":509,"line":761},[507,75755,75756],{"class":517},"    p0, p1 ",[507,75758,573],{"class":572},[507,75760,1616],{"class":517},[507,75762,59914],{"class":576},[507,75764,59917],{"class":517},[507,75766,75767],{"class":576},"trace",[507,75769,75770],{"class":517},"(s0)), np.",[507,75772,59914],{"class":576},[507,75774,59917],{"class":517},[507,75776,75767],{"class":576},[507,75778,75779],{"class":517},"(s1))\n",[507,75781,75782,75784,75787,75789,75792,75794,75797,75799],{"class":509,"line":775},[507,75783,1717],{"class":513},[507,75785,75786],{"class":517}," p0 ",[507,75788,5677],{"class":572},[507,75790,75791],{"class":517}," tol ",[507,75793,64173],{"class":513},[507,75795,75796],{"class":517}," p1 ",[507,75798,5677],{"class":572},[507,75800,75801],{"class":517}," tol:\n",[507,75803,75804,75806],{"class":509,"line":784},[507,75805,64873],{"class":513},[507,75807,58592],{"class":583},[507,75809,75810,75813,75815,75818,75820,75822,75824,75826,75828,75830,75832,75834,75837,75840,75842,75845,75847,75850],{"class":509,"line":796},[507,75811,75812],{"class":517},"    value ",[507,75814,573],{"class":572},[507,75816,75817],{"class":583}," 4.0",[507,75819,8229],{"class":572},[507,75821,75786],{"class":517},[507,75823,2391],{"class":572},[507,75825,75796],{"class":517},[507,75827,2391],{"class":572},[507,75829,58644],{"class":517},[507,75831,57927],{"class":583},[507,75833,65914],{"class":572},[507,75835,75836],{"class":576}," squared_fidelity",[507,75838,75839],{"class":517},"(s0 ",[507,75841,645],{"class":572},[507,75843,75844],{"class":517}," p0, s1 ",[507,75846,645],{"class":572},[507,75848,75849],{"class":517}," p1))",[507,75851,72708,75852],{"class":72706,"tabindex":72707},[507,75853,75854,75857],{"class":72711,"role":72712},[154,75855,75856],{},"The K score."," Read one qubit in the Z basis. K asks how distinguishable the other qubit's two conditional states become, weighted by how informative the readout was; the weight peaks at a 50\u002F50 split. Zero means the measurement reveals nothing about the partner.",[507,75859,75860,75862,75864,75866,75868,75871,75873,75875,75877],{"class":509,"line":809},[507,75861,2504],{"class":513},[507,75863,59620],{"class":572},[507,75865,59917],{"class":517},[507,75867,57917],{"class":576},[507,75869,75870],{"class":517},"(value, ",[507,75872,56714],{"class":583},[507,75874,622],{"class":517},[507,75876,57927],{"class":583},[507,75878,22540],{"class":517},[507,75880,75881],{"class":509,"line":1352},[507,75882,556],{"emptyLinePlaceholder":133},[507,75884,75885,75887,75890,75892,75894],{"class":509,"line":1357},[507,75886,1370],{"class":513},[507,75888,75889],{"class":576}," directed_pair_k",[507,75891,580],{"class":517},[507,75893,75688],{"class":1382},[507,75895,1883],{"class":517},[507,75897,75898],{"class":509,"line":1362},[507,75899,75900],{"class":730},"    \"\"\"(K[i->j], K[j->i]) from one 2-qubit reduced density matrix.\"\"\"\n",[507,75902,75903,75906,75908,75910,75913,75915,75917,75919,75922,75924,75926,75928,75931,75934],{"class":509,"line":1367},[507,75904,75905],{"class":517},"    k_ij ",[507,75907,573],{"class":572},[507,75909,75683],{"class":576},[507,75911,75912],{"class":517},"(rho2, [",[507,75914,601],{"class":583},[507,75916,622],{"class":517},[507,75918,625],{"class":583},[507,75920,75921],{"class":517},"], [",[507,75923,584],{"class":583},[507,75925,622],{"class":517},[507,75927,8226],{"class":583},[507,75929,75930],{"class":517},"])   ",[507,75932,75933],{"class":562},"# measure i, inspect j",[507,75935,72708,75936],{"class":72706,"tabindex":72707},[507,75937,75938,75941],{"class":72711,"role":72712},[154,75939,75940],{},"Direction matters."," The two calls swap which qubit is measured. K[i→j] and K[j→i] can genuinely differ, which undirected metrics like mutual information cannot express.",[507,75943,75944,75947,75949,75951,75953,75955,75957,75959,75961,75963,75965,75967,75969],{"class":509,"line":1379},[507,75945,75946],{"class":517},"    k_ji ",[507,75948,573],{"class":572},[507,75950,75683],{"class":576},[507,75952,75912],{"class":517},[507,75954,601],{"class":583},[507,75956,622],{"class":517},[507,75958,584],{"class":583},[507,75960,75921],{"class":517},[507,75962,625],{"class":583},[507,75964,622],{"class":517},[507,75966,8226],{"class":583},[507,75968,75930],{"class":517},[507,75970,75971],{"class":562},"# measure j, inspect i\n",[507,75973,75974,75976],{"class":509,"line":1389},[507,75975,2504],{"class":513},[507,75977,75978],{"class":517}," k_ij, k_ji\n",[507,75980,75981],{"class":509,"line":1397},[507,75982,556],{"emptyLinePlaceholder":133},[507,75984,75985],{"class":509,"line":1412},[507,75986,75987],{"class":562},"# The flow: truncate the circuit after every gate and rescore every pair.\n",[507,75989,75990,75992,75994,75996,75998,76000,76002,76004,76006,76009,76011,76013],{"class":509,"line":1431},[507,75991,1630],{"class":513},[507,75993,69270],{"class":517},[507,75995,1636],{"class":513},[507,75997,8221],{"class":572},[507,75999,580],{"class":517},[507,76001,1763],{"class":572},[507,76003,580],{"class":517},[507,76005,75554],{"class":583},[507,76007,76008],{"class":517},".data) ",[507,76010,2107],{"class":572},[507,76012,1426],{"class":583},[507,76014,1883],{"class":517},[507,76016,76017,76020,76022,76025,76027,76029,76032,76035],{"class":509,"line":1449},[507,76018,76019],{"class":517},"    state ",[507,76021,573],{"class":572},[507,76023,76024],{"class":576}," prefix_state",[507,76026,580],{"class":517},[507,76028,75554],{"class":583},[507,76030,76031],{"class":517},", m)   ",[507,76033,76034],{"class":562},"# statevector after the first m gates",[507,76036,72708,76037],{"class":72706,"tabindex":72707},[507,76038,76039,76042],{"class":72711,"role":72712},[154,76040,76041],{},"The flow."," Each marker is the circuit truncated after m gates. Rescoring at every marker turns a static diagram into motion: correlation appears at one CX, then gets handed down the line by the next.",[507,76044,76045,76047,76049,76051,76054,76056,76058,76060,76062,76064,76066,76069,76072,76075,76077,76079,76081,76083,76085],{"class":509,"line":1465},[507,76046,2060],{"class":572},[507,76048,580],{"class":517},[507,76050,22278],{"class":513},[507,76052,76053],{"class":730},"\"marker ",[507,76055,2810],{"class":583},[507,76057,4417],{"class":517},[507,76059,2872],{"class":583},[507,76061,24985],{"class":730},[507,76063,61723],{"class":572},[507,76065,2810],{"class":583},[507,76067,76068],{"class":576},"k_matrix",[507,76070,76071],{"class":517},"(state).",[507,76073,76074],{"class":576},"round",[507,76076,580],{"class":517},[507,76078,584],{"class":583},[507,76080,3649],{"class":517},[507,76082,2872],{"class":583},[507,76084,22281],{"class":730},[507,76086,587],{"class":517},[18,76088,76089,76090,76093,76094,76097],{},"The answer comes back directed. K",[507,76091,76092],{},"i→j"," and K",[507,76095,76096],{},"j→i"," can genuinely differ, so the qubit graph becomes a map with arrows rather than a symmetric mesh. And because every score is computed from two-qubit reduced density matrices, it stays cheap enough to drive the dashboard in real time.",[74616,76099,76101],{"lead":76100},"Three lenses on every pair of qubits, switchable per measurement basis.",[41852,76102,76103,76111],{},[41855,76104,76105],{},[41858,76106,76107,76109],{},[41861,76108,74627],{},[41861,76110,74630],{},[41868,76112,76113,76126,76134,76142],{},[41858,76114,76115,76118],{},[41873,76116,76117],{},"K-network",[41873,76119,76120,76121,76093,76123,76125],{},"The default: a directed, measurement-induced correlation score. K",[507,76122,76092],{},[507,76124,76096],{}," can differ.",[41858,76127,76128,76131],{},[41873,76129,76130],{},"Mutual information",[41873,76132,76133],{},"Undirected total correlation, classical and quantum together.",[41858,76135,76136,76139],{},[41873,76137,76138],{},"Entanglement of formation",[41873,76140,76141],{},"The entanglement on its own, separated from classical correlation.",[41858,76143,76144,76147],{},[41873,76145,76146],{},"Bases",[41873,76148,76149],{},"Every metric viewable under Z, X, and Y measurement.",[13,76151,76153],{"id":76152},"from-simulator-to-hardware","From simulator to hardware",[18,76155,76156],{},"Where do the density matrices come from? On a simulator, straight from the statevector. On hardware, the same numbers arrive by quantum state tomography: run the circuit many times, measure, and reconstruct each pair's state from the statistics. QCFlows treats the two as interchangeable sources feeding the same dashboard.",[18,76158,76159],{},"The experiment the team most wants to run on IonQ is a full tomography of a cat-state preparation, based on the algorithm in Iosifina's paper. Done faithfully, it needs mid-circuit measurement, a capability IonQ has slated for its upcoming Tempo processor. Until then, the team has a fallback ready.",[72443,76161,76164],{"avatar":104,"name":76162,"role":76163,"username":75404},"Dr. Iosifina Angelidi","Theory, QCFlows",[18,76165,76166],{},"Otherwise, we run each layer of unitaries and measurements, save the output state, and refit it into the next layer until we get the cat state. That is what the algorithm in the paper does.",[13,76168,76170],{"id":76169},"made-to-be-taken-apart","Made to be taken apart",[18,76172,76173],{},"The architecture is deliberately modular: a backend that computes quantum-information metrics and a frontend that visualizes them, talking over websockets, published as two separate repositories. The split is the point. You can run the full dashboard, or skip it and call the API from your own project, where every metric comes back as plain JSON.",[72443,76175,76176],{"avatar":104,"name":74407,"role":74408,"username":74409},[18,76177,76178],{},"If you just want to compute a mutual-information network metric, you should be able to take that part of the app and be happy about it.",[18,76180,76181],{},"The same thinking extends to hosting. The app lives at a public URL, but the repositories ship with instructions for running it yourself, and for Paulo that half of the offer matters more than the finished product.",[72443,76183,76184],{"avatar":104,"name":74407,"role":74408,"username":74409},[18,76185,76186],{},"Even more than giving a finished app, the contribution is to say: here are these metrics, these ways of looking into quantum algorithms. If you have an idea for a completely different application that uses this kind of data, you should be able to take the source code and deploy it yourself.",[13,76188,73027],{"id":73026},[18,76190,76191],{},"QCFlows is open and forkable on Qollab, both repositories are MIT-licensed, and the dashboard is live in your browser right now. Draw a few gates onto the qubit graph, or import a QASM file, and watch the correlation structure respond.",[73026,76193,76196],{"fork-href":76194,"live-href":75416,"title":76195},"\u002Fu\u002Fitaborala\u002Fqcflows","Draw a circuit. Watch it correlate.",[18,76197,76198,76199],{},"Fork QCFlows, load a circuit onto the qubit graph, and watch its correlation structure form and move, layer by layer. ",[154,76200,73040],{},[953,76202,76203],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":76205},[76206,76207,76208,76209,76210],{"id":75428,"depth":105,"text":75429},{"id":75492,"depth":105,"text":75493},{"id":76152,"depth":105,"text":76153},{"id":76169,"depth":105,"text":76170},{"id":73026,"depth":105,"text":73027},[112,969,74293],[76213,76224,76230],{"username":74409,"name":74407,"role":76214,"avatar":104,"bio":76215,"links":76216},"Project lead · quantum & music technology","Paulo is an interdisciplinary researcher working where physics meets music technology. A PhD student at the Cyprus Institute, part of the ERA-chair QUEST grant, in collaboration with DESY, he investigates variational quantum algorithms for high-energy physics and tools for sonifying and visualizing quantum computation. He also supports Quantum Patterns, a second project in this challenge.",[76217,76219,76222],{"label":73068,"href":76218},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fitaborala",{"label":76220,"href":76221},"Site ↗","https:\u002F\u002Fitabora.space\u002F",{"label":979,"href":76223},"https:\u002F\u002Fgithub.com\u002FItaborala",{"username":75404,"name":76162,"role":76225,"avatar":104,"bio":76226,"links":76227},"Theory","Iosifina is a postdoctoral research fellow at the Cyprus Institute, working on quantum circuits and entanglement stabilization. She leads the theory side of QCFlows, including the cat-state preparation the team plans to run as a full tomography experiment on IonQ hardware.",[76228],{"label":73068,"href":76229},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fiosifinaangelidi",{"username":75405,"name":75485,"role":76231,"avatar":104,"bio":76232,"links":76233},"Quantum information","Kostas is a researcher in the same QUEST group, focused on quantum information and hybrid algorithms for around a decade. He co-conceived QCFlows and works on its metrics and backend.",[76234],{"label":73068,"href":76235},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Feelvex",{"username":8,"name":73075,"role":73076,"avatar":73077},"Paulo Itaboraí, Iosifina Angelidi, and Kostas Blekos built a live dashboard that renders a quantum circuit as a dynamic graph of correlations, so you can watch entanglement form and move across the layers of an algorithm.","QCFlows renders a quantum circuit as a live graph of correlations, so you can watch entanglement form and move layer by layer. A Qollab Spring 2026 project.",{"href":76194,"label":76240},"Fork QCFlows",{"image":76242,"alt":76243,"liveUrl":75416},"\u002F_content\u002Fimages\u002Fqcflows\u002Fhero.webp","The QCFlows dashboard: a quantum circuit rendered as a dynamic qubit graph beside a correlation heatmap",{},"\u002Fblog\u002Fqcflows","2026-07-02",[],[76249,76252,76255],{"username":75023,"project":76250,"title":74564,"category":75052,"thumb":76251,"to":74563},"quantum-market-game","\u002F_content\u002Fimages\u002Fquantum-market-game\u002Fthumbnail.webp",{"username":74914,"project":76253,"title":74523,"category":75052,"thumb":76254,"to":74522},"quantum-systemic-oracle","\u002F_content\u002Fimages\u002Fsystemic-oracle\u002Fthumbnail.webp",{"username":72430,"project":73748,"title":73052,"category":1007,"thumb":73749,"to":73750},{"title":76257,"description":76258},"Quantum Creative Project Showcase: QCFlows","A quantum circuit as a dynamic graph of correlations: watch entanglement form and move, layer by layer.","blog\u002Fqcflows",[1019,143,74477],"VBTaF3A8WjQQ1_F-qGXswjaTgLCLD5cWGPQzIG2AUjE",{"id":76263,"title":76264,"authors":76265,"body":76266,"breadcrumb":76916,"builders":76917,"byline":76925,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":76926,"description":76927,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":76928,"hero":76930,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":76931,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":76932,"publishDate":76246,"readingTime":993,"related":76933,"relatedProjects":76934,"seo":76938,"stem":76941,"tags":76942,"track":116,"trackName":116,"__hash__":76943},"blog\u002Fblog\u002Fquantum-regime-radar.md","Project Showcase: Quantum Regime Radar",[74991],{"type":10,"value":76267,"toc":76910},[76268,76271,76274,76277,76282,76286,76289,76333,76336,76341,76345,76348,76825,76828,76875,76879,76882,76886,76889,76894,76896,76899,76907],[18,76269,76270],{},"A volatility model fitted in a calm market breaks in a crisis. Quantum Regime Radar asks the question a modeler needs answered first: which kind of market is this?",[18,76272,76273],{},"It classifies live equity returns against five volatility regimes taken from real market history, episodes like the 2017 melt-up or the buildup to the SVB collapse. Each regime's volatility fingerprint, estimated with the GARCH family of models, is encoded as a quantum state. At inference time a quantum kernel measures how strongly the live market's state overlaps each reference, and the result is a probability distribution over regimes, plus a recommendation for which modeling approach fits the moment.",[18,76275,76276],{},"It is a Spring 2026 challenge project from Alireza Khodaei, whose doctoral research put GARCH estimation on quantum hardware. His framing of the whole system fits in three sentences.",[72443,76278,76279],{"avatar":104,"name":74989,"role":74990,"username":74991},[18,76280,76281],{},"We have a bunch of known market regimes with respect to volatility, and we make a quantum fingerprint out of each. Then, given live data: how similar are these two days to the pre-crisis regime? There is no yes-or-no answer. There is a probability.",[13,76283,76285],{"id":76284},"five-regimes-taken-from-history","Five regimes, taken from history",[18,76287,76288],{},"The reference regimes are not synthetic labels. Each one is an empirically characterized market episode whose volatility behaviour was estimated and backtested with GARCH(1,1), the standard model that captures baseline variance, shock reactivity, and how long volatility persists. The current library holds five:",[74616,76290,76291],{"lead":74934},[41852,76292,76293,76301],{},[41855,76294,76295],{},[41858,76296,76297,76299],{},[41861,76298,74627],{},[41861,76300,74630],{},[41868,76302,76303,76309,76315,76321,76327],{},[41858,76304,76305,76307],{},[41873,76306,74951],{},[41873,76308,74954],{},[41858,76310,76311,76313],{},[41873,76312,74959],{},[41873,76314,74962],{},[41858,76316,76317,76319],{},[41873,76318,74967],{},[41873,76320,74970],{},[41858,76322,76323,76325],{},[41873,76324,74975],{},[41873,76326,74978],{},[41858,76328,76329,76331],{},[41873,76330,74983],{},[41873,76332,74986],{},[18,76334,76335],{},"Getting to five was work. The project started from twenty known regimes across the S&P 500, each fingerprinted with its own qubit budget per GARCH parameter. That full set was too large for today's hardware, so the team distilled it to the most principal regimes by entropy contribution and focused the live test cases on Magnificent 7 stocks, with baked-in windows like NVDA 2022 and META 2020.",[72443,76337,76338],{"avatar":104,"name":74989,"role":74990,"username":74991},[18,76339,76340],{},"With the original regime set we were over the capacity of the hardware. We needed 80 to 100 qubits to represent one regime. So we compressed the set to a leaner version, but we did not want to lose entropy. That was the challenge of the past few weeks.",[13,76342,76344],{"id":76343},"a-kernel-that-compares-histories","A kernel that compares histories",[18,76346,76347],{},"Each regime fingerprint is a probability distribution over the joint GARCH parameter space, and it is amplitude-encoded: the quantum state carries the square roots of those probabilities. That choice makes the physics do the statistics. The overlap between two encoded states works out to the Bhattacharyya coefficient between the two distributions, a similarity that compares where probability mass actually sits rather than the distance between point estimates. The playground cell ships with fitted twelve-qubit circuits baked in, so the whole pipeline runs self-contained.",[493,76349,76352],{"name":76350,"run-href":76351,"tag":72511},"regime_radar_cell.py","\u002Fu\u002Falireza\u002Fquantum-regime-radar",[498,76353,76355],{"className":500,"code":76354,"language":502,"meta":72515,"style":104},"# The playground cell: baked-in regime fingerprints, rebuilt as circuits,\n# scored by a quantum-kernel inversion test on the selected backend.\ndef _build(params, n=N_QUBITS, L=N_LAYERS):\n    \"\"\"Rebuild a fitted fingerprint state (12 qubits, 8 layers).\"\"\"\n    qc = QuantumCircuit(n)\n    pidx = 0\n    for q in range(n):\n        qc.ry(float(params[pidx]), q); pidx += 1\n    for _ in range(L):\n        for q in range(n - 1):\n            qc.cx(q, q + 1)\n        for q in range(n):\n            qc.ry(float(params[pidx]), q); pidx += 1\n    return qc\n\ndef _inversion(qc_live, qc_ref):\n    n = qc_live.num_qubits\n    qc = QuantumCircuit(n, n)\n    qc.compose(qc_live, inplace=True)\n    qc.compose(qc_ref.inverse(), inplace=True)\n    qc.measure(range(n), range(n))\n    return qc\n\n# Rank all five regimes offline first (exact overlaps from the baked\n# params), then spend the run's single hardware job verifying the top match.\nlive = _build(DEMO_PARAMS[ci][REF_FAM[target]])\nref  = _build(REF_PARAMS[target])\ncircuit = transpile(_inversion(live, ref), backend=backend, optimization_level=0)\ncounts  = backend.run(circuit, shots=shots).result().get_counts()\nK_meas  = _zero_frac(counts)\nprint(f\"Measured K = {K_meas:.4f} vs offline fitted K = {K_fit[target]:.4f}\")\n",[504,76356,76357,76362,76367,76400,76405,76415,76424,76437,76454,76467,76485,76508,76520,76536,76542,76546,76565,76574,76585,76602,76631,76649,76655,76659,76664,76669,76692,76709,76741,76768,76789],{"__ignoreMap":104},[507,76358,76359],{"class":509,"line":510},[507,76360,76361],{"class":562},"# The playground cell: baked-in regime fingerprints, rebuilt as circuits,\n",[507,76363,76364],{"class":509,"line":105},[507,76365,76366],{"class":562},"# scored by a quantum-kernel inversion test on the selected backend.\n",[507,76368,76369,76371,76374,76376,76379,76381,76383,76385,76388,76390,76393,76395,76398],{"class":509,"line":540},[507,76370,1370],{"class":513},[507,76372,76373],{"class":576}," _build",[507,76375,580],{"class":517},[507,76377,76378],{"class":1382},"params",[507,76380,622],{"class":517},[507,76382,4420],{"class":1382},[507,76384,573],{"class":517},[507,76386,76387],{"class":583},"N_QUBITS",[507,76389,622],{"class":517},[507,76391,76392],{"class":1382},"L",[507,76394,573],{"class":517},[507,76396,76397],{"class":583},"N_LAYERS",[507,76399,1883],{"class":517},[507,76401,76402],{"class":509,"line":553},[507,76403,76404],{"class":730},"    \"\"\"Rebuild a fitted fingerprint state (12 qubits, 8 layers).\"\"\"\n",[507,76406,76407,76409,76411,76413],{"class":509,"line":559},[507,76408,72833],{"class":517},[507,76410,573],{"class":572},[507,76412,577],{"class":576},[507,76414,64881],{"class":517},[507,76416,76417,76420,76422],{"class":509,"line":566},[507,76418,76419],{"class":517},"    pidx ",[507,76421,573],{"class":572},[507,76423,2246],{"class":583},[507,76425,76426,76428,76430,76432,76434],{"class":509,"line":590},[507,76427,1916],{"class":513},[507,76429,22114],{"class":517},[507,76431,1636],{"class":513},[507,76433,8221],{"class":572},[507,76435,76436],{"class":517},"(n):\n",[507,76438,76439,76441,76443,76445,76447,76450,76452],{"class":509,"line":610},[507,76440,72955],{"class":517},[507,76442,639],{"class":576},[507,76444,580],{"class":517},[507,76446,1406],{"class":572},[507,76448,76449],{"class":517},"(params[pidx]), q); pidx ",[507,76451,2285],{"class":572},[507,76453,2084],{"class":583},[507,76455,76456,76458,76460,76462,76464],{"class":509,"line":634},[507,76457,1916],{"class":513},[507,76459,8216],{"class":517},[507,76461,1636],{"class":513},[507,76463,8221],{"class":572},[507,76465,76466],{"class":517},"(L):\n",[507,76468,76469,76471,76473,76475,76477,76479,76481,76483],{"class":509,"line":661},[507,76470,2267],{"class":513},[507,76472,22114],{"class":517},[507,76474,1636],{"class":513},[507,76476,8221],{"class":572},[507,76478,68912],{"class":517},[507,76480,2367],{"class":572},[507,76482,1426],{"class":583},[507,76484,1883],{"class":517},[507,76486,76487,76489,76491,76494,76496,76498,76500],{"class":509,"line":678},[507,76488,72891],{"class":517},[507,76490,615],{"class":576},[507,76492,76493],{"class":517},"(q, q ",[507,76495,2107],{"class":572},[507,76497,1426],{"class":583},[507,76499,3649],{"class":517},[507,76501,72708,76502],{"class":72706,"tabindex":72707},[507,76503,76504,76507],{"class":72711,"role":72712},[154,76505,76506],{},"Entangling the register."," The CX ladder ties the qubit groups together, so the state can hold the regime's joint parameter structure (how shock reactivity couples to persistence) instead of three separate histograms.",[507,76509,76510,76512,76514,76516,76518],{"class":509,"line":683},[507,76511,2267],{"class":513},[507,76513,22114],{"class":517},[507,76515,1636],{"class":513},[507,76517,8221],{"class":572},[507,76519,76436],{"class":517},[507,76521,76522,76524,76526,76528,76530,76532,76534],{"class":509,"line":697},[507,76523,72891],{"class":517},[507,76525,639],{"class":576},[507,76527,580],{"class":517},[507,76529,1406],{"class":572},[507,76531,76449],{"class":517},[507,76533,2285],{"class":572},[507,76535,2084],{"class":583},[507,76537,76538,76540],{"class":509,"line":710},[507,76539,2504],{"class":513},[507,76541,72990],{"class":517},[507,76543,76544],{"class":509,"line":715},[507,76545,556],{"emptyLinePlaceholder":133},[507,76547,76548,76550,76553,76555,76558,76560,76563],{"class":509,"line":721},[507,76549,1370],{"class":513},[507,76551,76552],{"class":576}," _inversion",[507,76554,580],{"class":517},[507,76556,76557],{"class":1382},"qc_live",[507,76559,622],{"class":517},[507,76561,76562],{"class":1382},"qc_ref",[507,76564,1883],{"class":517},[507,76566,76567,76569,76571],{"class":509,"line":736},[507,76568,68277],{"class":517},[507,76570,573],{"class":572},[507,76572,76573],{"class":517}," qc_live.num_qubits\n",[507,76575,76576,76578,76580,76582],{"class":509,"line":748},[507,76577,72833],{"class":517},[507,76579,573],{"class":572},[507,76581,577],{"class":576},[507,76583,76584],{"class":517},"(n, n)\n",[507,76586,76587,76589,76591,76594,76596,76598,76600],{"class":509,"line":761},[507,76588,21867],{"class":517},[507,76590,13867],{"class":576},[507,76592,76593],{"class":517},"(qc_live, ",[507,76595,13873],{"class":2155},[507,76597,573],{"class":572},[507,76599,13878],{"class":583},[507,76601,587],{"class":517},[507,76603,76604,76606,76608,76611,76613,76615,76617,76619,76621,76623],{"class":509,"line":775},[507,76605,21867],{"class":517},[507,76607,13867],{"class":576},[507,76609,76610],{"class":517},"(qc_ref.",[507,76612,13825],{"class":576},[507,76614,13950],{"class":517},[507,76616,13873],{"class":2155},[507,76618,573],{"class":572},[507,76620,13878],{"class":583},[507,76622,3649],{"class":517},[507,76624,72708,76625],{"class":72706,"tabindex":72707},[507,76626,76627,76630],{"class":72711,"role":72712},[154,76628,76629],{},"The inversion test."," Prepare the live state, then run the reference circuit backwards. If the two states match, interference brings every amplitude back to the all-zeros outcome.",[507,76632,76633,76635,76637,76639,76641,76644,76646],{"class":509,"line":784},[507,76634,21867],{"class":517},[507,76636,72822],{"class":576},[507,76638,580],{"class":517},[507,76640,2204],{"class":572},[507,76642,76643],{"class":517},"(n), ",[507,76645,2204],{"class":572},[507,76647,76648],{"class":517},"(n))\n",[507,76650,76651,76653],{"class":509,"line":796},[507,76652,2504],{"class":513},[507,76654,72990],{"class":517},[507,76656,76657],{"class":509,"line":809},[507,76658,556],{"emptyLinePlaceholder":133},[507,76660,76661],{"class":509,"line":1352},[507,76662,76663],{"class":562},"# Rank all five regimes offline first (exact overlaps from the baked\n",[507,76665,76666],{"class":509,"line":1357},[507,76667,76668],{"class":562},"# params), then spend the run's single hardware job verifying the top match.\n",[507,76670,76671,76674,76676,76678,76680,76683,76686,76689],{"class":509,"line":1362},[507,76672,76673],{"class":517},"live ",[507,76675,573],{"class":572},[507,76677,76373],{"class":576},[507,76679,580],{"class":517},[507,76681,76682],{"class":583},"DEMO_PARAMS",[507,76684,76685],{"class":517},"[ci][",[507,76687,76688],{"class":583},"REF_FAM",[507,76690,76691],{"class":517},"[target]])\n",[507,76693,76694,76697,76699,76701,76703,76706],{"class":509,"line":1367},[507,76695,76696],{"class":517},"ref  ",[507,76698,573],{"class":572},[507,76700,76373],{"class":576},[507,76702,580],{"class":517},[507,76704,76705],{"class":583},"REF_PARAMS",[507,76707,76708],{"class":517},"[target])\n",[507,76710,76711,76713,76715,76718,76720,76723,76726,76728,76730,76732,76735,76737,76739],{"class":509,"line":1379},[507,76712,73339],{"class":517},[507,76714,573],{"class":572},[507,76716,76717],{"class":576}," transpile",[507,76719,580],{"class":517},[507,76721,76722],{"class":576},"_inversion",[507,76724,76725],{"class":517},"(live, ref), ",[507,76727,73507],{"class":2155},[507,76729,573],{"class":572},[507,76731,23943],{"class":517},[507,76733,76734],{"class":2155},"optimization_level",[507,76736,573],{"class":572},[507,76738,601],{"class":583},[507,76740,587],{"class":517},[507,76742,76743,76746,76748,76750,76752,76754,76756,76758,76760,76762,76764,76766],{"class":509,"line":1389},[507,76744,76745],{"class":517},"counts  ",[507,76747,573],{"class":572},[507,76749,73487],{"class":517},[507,76751,22501],{"class":576},[507,76753,73492],{"class":517},[507,76755,68762],{"class":2155},[507,76757,573],{"class":572},[507,76759,68812],{"class":517},[507,76761,23996],{"class":576},[507,76763,13983],{"class":517},[507,76765,73558],{"class":576},[507,76767,781],{"class":517},[507,76769,76770,76773,76775,76778,76781],{"class":509,"line":1397},[507,76771,76772],{"class":517},"K_meas  ",[507,76774,573],{"class":572},[507,76776,76777],{"class":576}," _zero_frac",[507,76779,76780],{"class":517},"(counts)",[507,76782,72708,76783],{"class":72706,"tabindex":72707},[507,76784,76785,76788],{"class":72711,"role":72712},[154,76786,76787],{},"The kernel is a probability."," The fraction of all-zeros shots estimates K, the squared overlap between the live and reference states. For these amplitude-encoded fingerprints, that equals the squared Bhattacharyya overlap between the two distributions.",[507,76790,76791,76793,76795,76797,76800,76802,76805,76807,76809,76812,76814,76817,76819,76821,76823],{"class":509,"line":1412},[507,76792,8525],{"class":572},[507,76794,580],{"class":517},[507,76796,22278],{"class":513},[507,76798,76799],{"class":730},"\"Measured K = ",[507,76801,2810],{"class":583},[507,76803,76804],{"class":517},"K_meas",[507,76806,70538],{"class":513},[507,76808,2872],{"class":583},[507,76810,76811],{"class":730}," vs offline fitted K = ",[507,76813,2810],{"class":583},[507,76815,76816],{"class":517},"K_fit[target]",[507,76818,70538],{"class":513},[507,76820,2872],{"class":583},[507,76822,22281],{"class":730},[507,76824,587],{"class":517},[18,76826,76827],{},"The playground permits one submitted job per run, so the cell works the way a careful experimentalist would: it prints the full five-regime ranking from exact offline overlaps first, then spends its single hardware job verifying one row of that ranking on the QPU, with shot-noise error bars on the measured kernel.",[74616,76829,76831],{"lead":76830},"One entry point, three ways to score the same kernel.",[41852,76832,76833,76841],{},[41855,76834,76835],{},[41858,76836,76837,76839],{},[41861,76838,74627],{},[41861,76840,74630],{},[41868,76842,76843,76851,76859,76867],{},[41858,76844,76845,76848],{},[41873,76846,76847],{},"Statevector",[41873,76849,76850],{},"Exact analytic overlap, no shots, for baselines and offline ranking.",[41858,76852,76853,76856],{},[41873,76854,76855],{},"Sampler",[41873,76857,76858],{},"Any Qiskit V2 sampler, from local simulation up to a real QPU.",[41858,76860,76861,76864],{},[41873,76862,76863],{},"IonQ",[41873,76865,76866],{},"Transpiled to the native gate set; simulator, Aria, or Forte hardware.",[41858,76868,76869,76872],{},[41873,76870,76871],{},"Circuits",[41873,76873,76874],{},"Inversion test on n qubits by default; a swap test when state preparation must stay a black box.",[13,76876,76878],{"id":76877},"honest-about-the-quantum","Honest about the quantum",[18,76880,76881],{},"Before claiming anything for the quantum side, the team built the classical case against themselves. A plain correlation study of the live windows against the regime library, run in NumPy, topped out around 0.4 on its best sample and sat far lower on the rest: too weak to use in finance. That baseline ships with the project, in one picture.",[72443,76883,76884],{"avatar":104,"name":74989,"role":74990,"username":74991},[18,76885,74994],{},[18,76887,76888],{},"The quantum claim is deliberately modest. The project does not promise an exponential speedup, and it concedes that a classical computer can evaluate the same similarity on a moderate histogram. The argument is about fit: amplitude-encoded states give the right inner product for comparing distributions by construction, entanglement carries the joint parameter structure without hand-engineered interaction features, and the same primitive runs unchanged from a statevector simulation to a QPU as the library grows. The open risk is equally plainly stated.",[72443,76890,76891],{"avatar":104,"name":74989,"role":74990,"username":74991},[18,76892,76893],{},"My concern is the noise level on real hardware. We are talking about entropy here. If the noise is large, it easily overlays the entropy of the signal, and we lose the signal.",[13,76895,73027],{"id":73026},[18,76897,76898],{},"The playground cell is self-contained: the regime library and five live fingerprints, including META 2020, AMZN 2020, NVDA 2022, AAPL 2021, and META 2022, are baked in, so it runs with no accounts and no data setup. Pick a ticker and a year, run the offline ranking, then point the same cell at an IonQ backend to verify a kernel on hardware.",[73026,76900,76902],{"fork-href":76351,"live-href":104,"title":76901},"Pick a market. Ask which history it rhymes with.",[18,76903,76904,76905],{},"Fork Quantum Regime Radar, score a live window against five regimes taken from real market history, and verify the top match on a QPU. ",[154,76906,73040],{},[953,76908,76909],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":76911},[76912,76913,76914,76915],{"id":76284,"depth":105,"text":76285},{"id":76343,"depth":105,"text":76344},{"id":76877,"depth":105,"text":76878},{"id":73026,"depth":105,"text":73027},[112,969,74544],[76918],{"username":74991,"name":76919,"role":76920,"avatar":104,"bio":76921,"links":76922},"Alireza Khodaei, PhD","Creator · quantum finance","Alireza holds a PhD in Computer Engineering and Computer Science from the University of Nebraska-Lincoln and an MBA with a finance specialization, and works at Nelnet. His doctoral work developed a quantum-enhanced framework for GARCH parameter estimation on a quantum annealer, backtested across market regimes, and that empirical foundation is what the Regime Radar's reference library is built on.",[76923],{"label":73068,"href":76924},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Falireza",{"username":8,"name":73075,"role":73076,"avatar":73077},"Alireza Khodaei built a tool that fingerprints live market behavior and scores it against five volatility regimes taken from real history, with quantum kernels on IonQ hardware.","Alireza Khodaei built Quantum Regime Radar: it scores live market returns against five volatility regimes from real history with quantum kernels on IonQ.",{"href":76351,"label":76929},"Fork Regime Radar",{"image":104,"alt":104,"liveUrl":104},{},"\u002Fblog\u002Fquantum-regime-radar",[],[76935,76936,76937],{"username":73742,"project":73743,"title":73744,"category":73097,"thumb":73745,"to":73746},{"username":72430,"project":73748,"title":73052,"category":1007,"thumb":73749,"to":73750},{"username":73101,"project":73102,"title":73103,"category":73097,"thumb":73104,"to":73105},{"title":76939,"description":76940},"Quantum Creative Project Showcase: Quantum Regime Radar","Which kind of market is this? Live returns scored against five historical volatility regimes by quantum-kernel overlap on IonQ.","blog\u002Fquantum-regime-radar",[75067,143,75069],"Y8m0a3YugXi8HujCSOW7T7IqiBwEhIvQkvyUVjl3VfU",{"id":76945,"title":76946,"authors":76947,"body":76948,"breadcrumb":77779,"builders":77780,"byline":77792,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":77793,"description":77794,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":77795,"hero":77797,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":77800,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":77801,"publishDate":77802,"readingTime":74132,"related":77803,"relatedProjects":77804,"seo":77812,"stem":77815,"tags":77816,"track":116,"trackName":116,"__hash__":77817},"blog\u002Fblog\u002Fquantum-advantage-lab.md","Project Showcase: Quantum Advantage Lab",[74372],{"type":10,"value":76949,"toc":77773},[76950,76953,76956,76961,76965,76968,76974,76980,76986,76992,76997,77001,77004,77008,77011,77014,77018,77021,77701,77704,77756,77758,77761,77770],[18,76951,76952],{},"Most people are told that quantum computers are faster. Far fewer ever see why. Quantum Advantage Lab is built to close that gap: pick one of four famous algorithms, press go, and watch it run step by step beside the classical method solving the same problem.",[18,76954,76955],{},"The four races span the canon: Grover's search, a variational eigensolver for molecular ground states, a discrete-time quantum walk, and Hamiltonian simulation. Each one streams the quantum computation's intermediate state, the amplitudes climbing, the energy converging, the probability spreading, next to a classical baseline doing the same job. It is built and MIT-licensed by Hossein Sadeghi, a quantum-software veteran with a decade at D-Wave and Pasqal, and it runs on IonQ's trapped-ion hardware. What makes it unusual is not that it claims a win. It is how honest it is about what winning even means.",[72443,76957,76958],{"avatar":104,"name":74272,"role":74371,"username":74372},[18,76959,76960],{},"The whole idea started from the race: what algorithms can we show, and how do we visualize that competition? I had written the proposal, but it was not a case of \"I know the answer, I just need to implement it.\" It was going to be challenging.",[13,76962,76964],{"id":76963},"the-four-races","The four races",[18,76966,76967],{},"All four races share a shape. On one side, a quantum circuit; on the other, the best classical method for the same task. Both run, and the Lab streams what is happening inside each, not just the answer at the end.",[18,76969,76970,76973],{},[154,76971,76972],{},"Grover's search"," races amplitude amplification against a brute-force scan. You watch the amplitude of the target answer climb in a smooth arc while the classical search checks items one at a time. The picture makes Grover's true nature obvious: it is less a search than a rotation, one you can over-shoot if you run it too long.",[18,76975,76976,76979],{},[154,76977,76978],{},"VQE",", the variational quantum eigensolver, hunts for a molecule's ground-state energy. You watch the energy descend toward the exact value, with a chemical-accuracy band drawn in, while a classical optimizer works the same landscape. It is the one race where the quantum side is genuinely hard, and the Lab shows you why: the optimization landscape is full of traps.",[18,76981,76982,76985],{},[154,76983,76984],{},"The quantum walk"," sets a coined walk loose against an ordinary random walk on the same graph. Interference makes the quantum distribution spread ballistically, with sharp peaks at its edges, while the classical walk just diffuses into a bell curve. The gap between spreading like the square root of time and spreading linearly with time is the whole story, drawn live.",[18,76987,76988,76991],{},[154,76989,76990],{},"Hamiltonian simulation"," evolves a spin chain with a Trotterized circuit and races it against direct matrix exponentiation. Finer time-slices mean a more faithful result and a deeper circuit, and you watch that depth-versus-accuracy tradeoff play out as the fidelity climbs. This is the race that ships ready to run in the code below.",[72443,76993,76994],{"avatar":104,"name":74272,"role":74371,"username":74372},[18,76995,76996],{},"It does not just show static circuits or final answers. It runs real Qiskit circuits, streams the intermediate solver state, pairs each quantum method with a meaningful classical baseline, and is shaped around IonQ-native execution.",[13,76998,77000],{"id":76999},"an-honest-race","An honest race",[18,77002,77003],{},"Here is the part that most \"watch quantum win\" demos quietly skip. At the sizes that run on today's hardware, the quantum side does not finish first on a stopwatch. Four qubits are trivial for a laptop, every gate carries noise, and real jobs wait in a queue behind everyone else's. Quantum Advantage Lab does not pretend otherwise.",[72443,77005,77006],{"avatar":104,"name":74272,"role":74371,"username":74372},[18,77007,74375],{},[18,77009,77010],{},"So the Lab races the right thing. Not wall-clock time, but the mechanism: how many steps each method needs, how the quantum state evolves, where interference or amplitude amplification does its work. That is where the asymptotic story actually lives, and it is visible long before any hardware is genuinely faster.",[18,77012,77013],{},"It is also honest about the hardware itself. Live runs on IonQ go through a real queue and come back with real noise, so the Lab captures genuine hardware results and replays them on demand, each one labeled for what it is, with a clean statevector simulator alongside as the \"this is what perfect looks like\" reference. And because it targets IonQ's trapped-ion processor, the entangling layers in Grover and VQE map onto all-to-all connectivity without the SWAP overhead a superconducting chip would pay. The Lab can show you that directly, by comparing the transpiled circuits side by side.",[13,77015,77017],{"id":77016},"how-it-works","How it works",[18,77019,77020],{},"Every race is a real circuit, not an animation. The Hamiltonian-simulation race is the most self-contained, and it ships ready to run in the Qollab Playground. It builds a transverse-field Ising chain, evolves it with a first-order Trotter circuit, and measures how close the sampled result is to exact evolution as the number of Trotter steps climbs:",[493,77022,77025],{"name":77023,"run-href":77024,"tag":72511},"hamiltonian_race.py","\u002Fu\u002Fhsadeghi\u002Fquantum-advantage-lab",[498,77026,77028],{"className":500,"code":77027,"language":502,"meta":72515,"style":104},"# Quantum Advantage Lab: the Hamiltonian Simulation race.\n# Evolve a transverse-field Ising chain with a Trotter circuit, then sweep\n# the step count and watch the quantum result close in on exact evolution.\nimport numpy as np\nfrom scipy.linalg import expm\nfrom qiskit import QuantumCircuit, transpile\nfrom qiskit.circuit.library import PauliEvolutionGate\nfrom qiskit.quantum_info import SparsePauliOp\nfrom qiskit.synthesis import LieTrotter\n\nN_QUBITS, TIME = 4, 0.5\nN_STEPS_SWEEP  = [1, 2, 4, 8, 16]\n\ndef build_ising(n, J=1.0, h=1.0):              # H = -J sum ZZ - h sum X\n    terms = []\n    for i in range(n - 1):\n        zz = [\"I\"] * n; zz[i] = zz[i + 1] = \"Z\"\n        terms.append((\"\".join(zz), -J))\n    for i in range(n):\n        x = [\"I\"] * n; x[i] = \"X\"\n        terms.append((\"\".join(x), -h))\n    return SparsePauliOp.from_list(terms)\n\ndef trotter_circuit(H, t, n_steps, n):         # first-order Lie-Trotter\n    qc = QuantumCircuit(n, n)\n    qc.append(PauliEvolutionGate(H, time=t, synthesis=LieTrotter(reps=n_steps)), range(n))\n    qc.measure(range(n), range(n))\n    return qc\n\nH     = build_ising(N_QUBITS)\nexact = exact_distribution(H, TIME, N_QUBITS)   # classical baseline, via SciPy expm\n\nfor n_steps in N_STEPS_SWEEP:\n    qc     = trotter_circuit(H, TIME, n_steps, N_QUBITS)\n    tqc    = transpile(qc, backend, optimization_level=1)   # IonQ-native gates\n    counts = backend.run(tqc, shots=shots).result().get_counts()\n    probs  = counts_to_probs(counts, N_QUBITS)\n    print(f\"steps={n_steps:>2}  depth={tqc.depth():>3}  TV(quantum, exact)={tv_distance(probs, exact):.4f}\")\n",[504,77029,77030,77035,77040,77045,77055,77067,77078,77088,77099,77109,77113,77131,77162,77166,77199,77208,77226,77259,77282,77294,77316,77338,77357,77361,77391,77401,77452,77468,77474,77478,77493,77524,77528,77542,77562,77593,77620,77637],{"__ignoreMap":104},[507,77031,77032],{"class":509,"line":510},[507,77033,77034],{"class":562},"# Quantum Advantage Lab: the Hamiltonian Simulation race.\n",[507,77036,77037],{"class":509,"line":105},[507,77038,77039],{"class":562},"# Evolve a transverse-field Ising chain with a Trotter circuit, then sweep\n",[507,77041,77042],{"class":509,"line":540},[507,77043,77044],{"class":562},"# the step count and watch the quantum result close in on exact evolution.\n",[507,77046,77047,77049,77051,77053],{"class":509,"line":553},[507,77048,514],{"class":513},[507,77050,518],{"class":517},[507,77052,521],{"class":513},[507,77054,524],{"class":517},[507,77056,77057,77059,77062,77064],{"class":509,"line":559},[507,77058,529],{"class":513},[507,77060,77061],{"class":517}," scipy.linalg ",[507,77063,514],{"class":513},[507,77065,77066],{"class":517}," expm\n",[507,77068,77069,77071,77073,77075],{"class":509,"line":566},[507,77070,529],{"class":513},[507,77072,532],{"class":517},[507,77074,514],{"class":513},[507,77076,77077],{"class":517}," QuantumCircuit, transpile\n",[507,77079,77080,77082,77084,77086],{"class":509,"line":590},[507,77081,529],{"class":513},[507,77083,1273],{"class":517},[507,77085,514],{"class":513},[507,77087,1278],{"class":517},[507,77089,77090,77092,77094,77096],{"class":509,"line":610},[507,77091,529],{"class":513},[507,77093,545],{"class":517},[507,77095,514],{"class":513},[507,77097,77098],{"class":517}," SparsePauliOp\n",[507,77100,77101,77103,77105,77107],{"class":509,"line":634},[507,77102,529],{"class":513},[507,77104,1285],{"class":517},[507,77106,514],{"class":513},[507,77108,1290],{"class":517},[507,77110,77111],{"class":509,"line":661},[507,77112,556],{"emptyLinePlaceholder":133},[507,77114,77115,77117,77119,77122,77124,77126,77128],{"class":509,"line":678},[507,77116,76387],{"class":583},[507,77118,622],{"class":517},[507,77120,77121],{"class":583},"TIME",[507,77123,1423],{"class":572},[507,77125,648],{"class":583},[507,77127,622],{"class":517},[507,77129,77130],{"class":583},"0.5\n",[507,77132,77133,77136,77139,77141,77143,77145,77147,77149,77151,77153,77155,77157,77160],{"class":509,"line":683},[507,77134,77135],{"class":583},"N_STEPS_SWEEP",[507,77137,77138],{"class":572},"  =",[507,77140,8427],{"class":517},[507,77142,625],{"class":583},[507,77144,622],{"class":517},[507,77146,584],{"class":583},[507,77148,622],{"class":517},[507,77150,12152],{"class":583},[507,77152,622],{"class":517},[507,77154,35740],{"class":583},[507,77156,622],{"class":517},[507,77158,77159],{"class":583},"16",[507,77161,1794],{"class":517},[507,77163,77164],{"class":509,"line":697},[507,77165,556],{"emptyLinePlaceholder":133},[507,77167,77168,77170,77173,77175,77177,77179,77181,77183,77185,77187,77189,77191,77193,77196],{"class":509,"line":710},[507,77169,1370],{"class":513},[507,77171,77172],{"class":576}," build_ising",[507,77174,580],{"class":517},[507,77176,4420],{"class":1382},[507,77178,622],{"class":517},[507,77180,7759],{"class":1382},[507,77182,573],{"class":517},[507,77184,57927],{"class":583},[507,77186,622],{"class":517},[507,77188,596],{"class":1382},[507,77190,573],{"class":517},[507,77192,57927],{"class":583},[507,77194,77195],{"class":517},"):              ",[507,77197,77198],{"class":562},"# H = -J sum ZZ - h sum X\n",[507,77200,77201,77204,77206],{"class":509,"line":715},[507,77202,77203],{"class":517},"    terms ",[507,77205,573],{"class":572},[507,77207,1910],{"class":517},[507,77209,77210,77212,77214,77216,77218,77220,77222,77224],{"class":509,"line":721},[507,77211,1916],{"class":513},[507,77213,8246],{"class":517},[507,77215,1636],{"class":513},[507,77217,8221],{"class":572},[507,77219,68912],{"class":517},[507,77221,2367],{"class":572},[507,77223,1426],{"class":583},[507,77225,1883],{"class":517},[507,77227,77228,77231,77233,77235,77237,77239,77241,77244,77246,77249,77251,77253,77255,77257],{"class":509,"line":736},[507,77229,77230],{"class":517},"        zz ",[507,77232,573],{"class":572},[507,77234,8427],{"class":517},[507,77236,8203],{"class":730},[507,77238,8206],{"class":517},[507,77240,2391],{"class":572},[507,77242,77243],{"class":517}," n; zz[i] ",[507,77245,573],{"class":572},[507,77247,77248],{"class":517}," zz[i ",[507,77250,2107],{"class":572},[507,77252,1426],{"class":583},[507,77254,8206],{"class":517},[507,77256,573],{"class":572},[507,77258,8268],{"class":730},[507,77260,77261,77264,77266,77268,77270,77272,77274,77277,77279],{"class":509,"line":748},[507,77262,77263],{"class":517},"        terms.",[507,77265,1939],{"class":576},[507,77267,63480],{"class":517},[507,77269,8430],{"class":730},[507,77271,53],{"class":517},[507,77273,8435],{"class":576},[507,77275,77276],{"class":517},"(zz), ",[507,77278,2367],{"class":572},[507,77280,77281],{"class":517},"J))\n",[507,77283,77284,77286,77288,77290,77292],{"class":509,"line":761},[507,77285,1916],{"class":513},[507,77287,8246],{"class":517},[507,77289,1636],{"class":513},[507,77291,8221],{"class":572},[507,77293,76436],{"class":517},[507,77295,77296,77299,77301,77303,77305,77307,77309,77312,77314],{"class":509,"line":775},[507,77297,77298],{"class":517},"        x ",[507,77300,573],{"class":572},[507,77302,8427],{"class":517},[507,77304,8203],{"class":730},[507,77306,8206],{"class":517},[507,77308,2391],{"class":572},[507,77310,77311],{"class":517}," n; x[i] ",[507,77313,573],{"class":572},[507,77315,8321],{"class":730},[507,77317,77318,77320,77322,77324,77326,77328,77330,77333,77335],{"class":509,"line":784},[507,77319,77263],{"class":517},[507,77321,1939],{"class":576},[507,77323,63480],{"class":517},[507,77325,8430],{"class":730},[507,77327,53],{"class":517},[507,77329,8435],{"class":576},[507,77331,77332],{"class":517},"(x), ",[507,77334,2367],{"class":572},[507,77336,77337],{"class":517},"h))\n",[507,77339,77340,77342,77344,77346,77349],{"class":509,"line":796},[507,77341,2504],{"class":513},[507,77343,8514],{"class":517},[507,77345,8517],{"class":576},[507,77347,77348],{"class":517},"(terms)",[507,77350,72708,77351],{"class":72706,"tabindex":72707},[507,77352,77353,77356],{"class":72711,"role":72712},[154,77354,77355],{},"The model."," A 1D transverse-field Ising chain: neighbouring spins coupled along Z, a field along X. A small, well-understood system to simulate.",[507,77358,77359],{"class":509,"line":809},[507,77360,556],{"emptyLinePlaceholder":133},[507,77362,77363,77365,77368,77370,77372,77374,77376,77378,77381,77383,77385,77388],{"class":509,"line":1352},[507,77364,1370],{"class":513},[507,77366,77367],{"class":576}," trotter_circuit",[507,77369,580],{"class":517},[507,77371,3138],{"class":1382},[507,77373,622],{"class":517},[507,77375,3298],{"class":1382},[507,77377,622],{"class":517},[507,77379,77380],{"class":1382},"n_steps",[507,77382,622],{"class":517},[507,77384,4420],{"class":1382},[507,77386,77387],{"class":517},"):         ",[507,77389,77390],{"class":562},"# first-order Lie-Trotter\n",[507,77392,77393,77395,77397,77399],{"class":509,"line":1357},[507,77394,72833],{"class":517},[507,77396,573],{"class":572},[507,77398,577],{"class":576},[507,77400,76584],{"class":517},[507,77402,77403,77405,77407,77409,77412,77415,77417,77419,77421,77423,77425,77427,77429,77431,77433,77436,77438,77441],{"class":509,"line":1362},[507,77404,21867],{"class":517},[507,77406,1939],{"class":576},[507,77408,580],{"class":517},[507,77410,77411],{"class":576},"PauliEvolutionGate",[507,77413,77414],{"class":517},"(H, ",[507,77416,9348],{"class":2155},[507,77418,573],{"class":572},[507,77420,9353],{"class":517},[507,77422,9356],{"class":2155},[507,77424,573],{"class":572},[507,77426,9361],{"class":576},[507,77428,580],{"class":517},[507,77430,9366],{"class":2155},[507,77432,573],{"class":572},[507,77434,77435],{"class":517},"n_steps)), ",[507,77437,2204],{"class":572},[507,77439,77440],{"class":517},"(n))",[507,77442,72708,77443],{"class":72706,"tabindex":72707},[507,77444,77445,77448,77449,77451],{"class":72711,"role":72712},[154,77446,77447],{},"Trotterization."," Approximates the time-evolution by chopping it into ",[504,77450,77380],{}," slices. More steps means a more faithful result and a deeper circuit, and watching that tradeoff is the race.",[507,77453,77454,77456,77458,77460,77462,77464,77466],{"class":509,"line":1367},[507,77455,21867],{"class":517},[507,77457,72822],{"class":576},[507,77459,580],{"class":517},[507,77461,2204],{"class":572},[507,77463,76643],{"class":517},[507,77465,2204],{"class":572},[507,77467,76648],{"class":517},[507,77469,77470,77472],{"class":509,"line":1379},[507,77471,2504],{"class":513},[507,77473,72990],{"class":517},[507,77475,77476],{"class":509,"line":1389},[507,77477,556],{"emptyLinePlaceholder":133},[507,77479,77480,77483,77485,77487,77489,77491],{"class":509,"line":1397},[507,77481,77482],{"class":517},"H     ",[507,77484,573],{"class":572},[507,77486,77172],{"class":576},[507,77488,580],{"class":517},[507,77490,76387],{"class":583},[507,77492,587],{"class":517},[507,77494,77495,77498,77500,77503,77505,77507,77509,77511,77513,77516],{"class":509,"line":1412},[507,77496,77497],{"class":517},"exact ",[507,77499,573],{"class":572},[507,77501,77502],{"class":576}," exact_distribution",[507,77504,77414],{"class":517},[507,77506,77121],{"class":583},[507,77508,622],{"class":517},[507,77510,76387],{"class":583},[507,77512,67189],{"class":517},[507,77514,77515],{"class":562},"# classical baseline, via SciPy expm",[507,77517,72708,77518],{"class":72706,"tabindex":72707},[507,77519,77520,77523],{"class":72711,"role":72712},[154,77521,77522],{},"The classical side."," SciPy exponentiates the full 2ⁿ×2ⁿ matrix directly. Exact, but the cost explodes with every qubit you add.",[507,77525,77526],{"class":509,"line":1431},[507,77527,556],{"emptyLinePlaceholder":133},[507,77529,77530,77532,77535,77537,77540],{"class":509,"line":1449},[507,77531,1630],{"class":513},[507,77533,77534],{"class":517}," n_steps ",[507,77536,1636],{"class":513},[507,77538,77539],{"class":583}," N_STEPS_SWEEP",[507,77541,1728],{"class":517},[507,77543,77544,77547,77549,77551,77553,77555,77558,77560],{"class":509,"line":1465},[507,77545,77546],{"class":517},"    qc     ",[507,77548,573],{"class":572},[507,77550,77367],{"class":576},[507,77552,77414],{"class":517},[507,77554,77121],{"class":583},[507,77556,77557],{"class":517},", n_steps, ",[507,77559,76387],{"class":583},[507,77561,587],{"class":517},[507,77563,77564,77567,77569,77571,77574,77576,77578,77580,77582,77585],{"class":509,"line":1471},[507,77565,77566],{"class":517},"    tqc    ",[507,77568,573],{"class":572},[507,77570,76717],{"class":576},[507,77572,77573],{"class":517},"(qc, backend, ",[507,77575,76734],{"class":2155},[507,77577,573],{"class":572},[507,77579,625],{"class":583},[507,77581,67189],{"class":517},[507,77583,77584],{"class":562},"# IonQ-native gates",[507,77586,72708,77587],{"class":72706,"tabindex":72707},[507,77588,77589,77592],{"class":72711,"role":72712},[154,77590,77591],{},"IonQ-native."," On trapped-ion hardware, all-to-all connectivity lets the entangling layers run with no SWAP gates, so the circuit stays shallow. Switch the backend to compare.",[507,77594,77595,77597,77599,77601,77603,77606,77608,77610,77612,77614,77616,77618],{"class":509,"line":1477},[507,77596,73551],{"class":517},[507,77598,573],{"class":572},[507,77600,73487],{"class":517},[507,77602,22501],{"class":576},[507,77604,77605],{"class":517},"(tqc, ",[507,77607,68762],{"class":2155},[507,77609,573],{"class":572},[507,77611,68812],{"class":517},[507,77613,23996],{"class":576},[507,77615,13983],{"class":517},[507,77617,73558],{"class":576},[507,77619,781],{"class":517},[507,77621,77622,77625,77627,77630,77633,77635],{"class":509,"line":1482},[507,77623,77624],{"class":517},"    probs  ",[507,77626,573],{"class":572},[507,77628,77629],{"class":576}," counts_to_probs",[507,77631,77632],{"class":517},"(counts, ",[507,77634,76387],{"class":583},[507,77636,587],{"class":517},[507,77638,77639,77641,77643,77645,77648,77650,77652,77655,77657,77660,77662,77665,77667,77669,77672,77674,77677,77679,77682,77685,77687,77689,77691,77693],{"class":509,"line":1488},[507,77640,2060],{"class":572},[507,77642,580],{"class":517},[507,77644,22278],{"class":513},[507,77646,77647],{"class":730},"\"steps=",[507,77649,2810],{"class":583},[507,77651,77380],{"class":517},[507,77653,77654],{"class":513},":>2",[507,77656,2872],{"class":583},[507,77658,77659],{"class":730},"  depth=",[507,77661,2810],{"class":583},[507,77663,77664],{"class":517},"tqc.",[507,77666,14179],{"class":576},[507,77668,66172],{"class":517},[507,77670,77671],{"class":513},":>3",[507,77673,2872],{"class":583},[507,77675,77676],{"class":730},"  TV(quantum, exact)=",[507,77678,2810],{"class":583},[507,77680,77681],{"class":576},"tv_distance",[507,77683,77684],{"class":517},"(probs, exact)",[507,77686,70538],{"class":513},[507,77688,2872],{"class":583},[507,77690,22281],{"class":730},[507,77692,3649],{"class":517},[507,77694,72708,77695],{"class":72706,"tabindex":72707},[507,77696,77697,77700],{"class":72711,"role":72712},[154,77698,77699],{},"The verdict."," Total-variation distance between the sampled quantum distribution and the exact one. Watch it shrink as the Trotter steps climb.",[18,77702,77703],{},"The same pattern drives the other three races: a real circuit on one side, an exact or best-effort classical solver on the other, and a stream of intermediate state in between. Because the architecture is modular, each race is a self-contained plug-in, which is what lets the Lab grow a fifth or sixth race without a rewrite.",[74616,77705,77707],{"lead":77706},"Quantum Advantage Lab is open source and MIT-licensed, with a modular architecture built for community-contributed races.",[41852,77708,77709,77717],{},[41855,77710,77711],{},[41858,77712,77713,77715],{},[41861,77714,74627],{},[41861,77716,74630],{},[41868,77718,77719,77726,77733,77741,77748],{},[41858,77720,77721,77723],{},[41873,77722,72215],{},[41873,77724,77725],{},"Qiskit, Grover, VQE, a quantum walk, and Hamiltonian simulation, each a real circuit.",[41858,77727,77728,77730],{},[41873,77729,72393],{},[41873,77731,77732],{},"qiskit-ionq, IonQ Forte trapped-ion QPU, all-to-all connectivity.",[41858,77734,77735,77738],{},[41873,77736,77737],{},"Classical",[41873,77739,77740],{},"NumPy, SciPy, tensor-network baselines, the side each race has to beat.",[41858,77742,77743,77745],{},[41873,77744,73859],{},[41873,77746,77747],{},"Streaming intermediate state, amplitudes, energies, distributions, and fidelity, step by step.",[41858,77749,77750,77753],{},[41873,77751,77752],{},"License",[41873,77754,77755],{},"MIT, with a plug-in architecture for community-contributed modules.",[13,77757,73027],{"id":73026},[18,77759,77760],{},"Quantum Advantage Lab is open and forkable on Qollab, MIT-licensed on GitHub, and live on the web right now. Pick a race, set the parameters, and step through it on a simulator or a real IonQ processor. The architecture is modular, so a new race is a plug-in, not a rewrite.",[73026,77762,77765],{"fork-href":77024,"live-href":77763,"title":77764},"https:\u002F\u002Fquantum-advantage-lab.vercel.app\u002F","Watch the speedup, and where it runs out.",[18,77766,77767,77768],{},"Fork the Lab, choose Grover, VQE, a quantum walk, or Hamiltonian simulation, and step through it beside its classical rival. ",[154,77769,73040],{},[953,77771,77772],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":77774},[77775,77776,77777,77778],{"id":76963,"depth":105,"text":76964},{"id":76999,"depth":105,"text":77000},{"id":77016,"depth":105,"text":77017},{"id":73026,"depth":105,"text":73027},[112,969,74261],[77781],{"username":74372,"name":77782,"role":77783,"avatar":104,"bio":77784,"links":77785},"Hossein Sadeghi Esfahani","Creator · quantum software & hardware","Hossein holds a PhD from the University of British Columbia and has spent more than a decade in quantum software. He spent seven years at D-Wave Systems, rising from applied researcher to solution architect and team lead and co-inventing three patents in quantum optimization and benchmarking, then led academic and R&D partnerships on neutral-atom systems at Pasqal Canada. He has also served as an investigator in the Creative Destruction Lab's Quantum Stream, mentoring early-stage quantum startups. Quantum Advantage Lab is his solo build.",[77786,77788,77790],{"label":73068,"href":77787},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fhsadeghi",{"label":73059,"href":77789},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fhosseinsadeghi\u002F",{"label":979,"href":77791},"https:\u002F\u002Fgithub.com\u002Fhosseinsadeghi",{"username":8,"name":73075,"role":73076,"avatar":73077},"Hossein Sadeghi built a lab where four quantum algorithms run beside their classical counterparts and stream their intermediate state, so the mechanism behind a speedup is something you watch unfold rather than read about.","Quantum Advantage Lab runs four quantum algorithms beside their classical counterparts, streaming each step so the mechanism behind a speedup is something you watch. A Qollab Spring 2026 project.",{"href":77024,"label":77796},"Fork the Lab",{"image":77798,"alt":77799,"liveUrl":77763},"\u002F_content\u002Fimages\u002Fquantum-advantage-lab\u002Fhero.webp","Quantum Advantage Lab's Race view: a quantum solver and a classical solver running the same Hamiltonian simulation side by side, state distributions streaming step by step",{},"\u002Fblog\u002Fquantum-advantage-lab","2026-06-30",[],[77805,77808,77809],{"username":74390,"project":77806,"title":74277,"category":73752,"thumb":77807,"to":74383},"qorbital","\u002F_content\u002Fimages\u002Fqorbital\u002Fthumbnail.webp",{"username":1011,"project":1012,"title":1013,"category":1007,"thumb":1014,"to":73092},{"username":74336,"project":77810,"title":74228,"category":73752,"thumb":77811,"to":74329},"quantum-canvas","\u002F_content\u002Fimages\u002Fquantum-canvas\u002Fthumbnail.webp",{"title":77813,"description":77814},"Quantum Creative Project Showcase: Quantum Advantage Lab","Four quantum algorithms run beside their classical counterparts, streaming each step, so the mechanism behind a speedup becomes something you can watch.","blog\u002Fquantum-advantage-lab",[74477,143,1019],"E7hA8Ne2mFhtYetTHyhZAz6QcGv9aGM37odhQgwaM2U",{"id":77819,"title":77820,"authors":77821,"body":77822,"breadcrumb":78386,"builders":78387,"byline":78404,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":78405,"description":78406,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":78407,"hero":78408,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":78409,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":78410,"publishDate":77802,"readingTime":993,"related":78411,"relatedProjects":78412,"seo":78416,"stem":78419,"tags":78420,"track":116,"trackName":116,"__hash__":78422},"blog\u002Fblog\u002Fquantum-patterns.md","Project Showcase: Quantum Patterns",[73742,74409],{"type":10,"value":77823,"toc":78379},[77824,77827,77830,77836,77840,77844,77847,77850,77889,77894,77898,77905,77909,78280,78283,78328,78332,78335,78338,78342,78346,78349,78354,78357,78361,78363,78366,78376],[18,77825,77826],{},"Quantum Patterns starts from a provocation: what if the raw material of a piece of music came from a quantum circuit, from patterns that are coherent but never repeat, and that no classical computer could fake?",[18,77828,77829],{},"It is a browser-based live-coding instrument. Partitioned quantum cellular automata (PQCA) are evolved on a quantum circuit, their measurement outcomes captured as datasets, and those datasets loaded into a fork of Satori, where you write short scripts in plain JavaScript to turn them into pitch, rhythm, timbre, texture, and space. No Python, no accounts, and no quantum background needed to start.",[18,77831,77832,77833,77835],{},"It is a Spring 2026 challenge project led by Peter Thomas, who performs as ",[1031,77834,75305],{},", with Paulo Itaboraí on the quantum side. Peter comes to it from a doctorate spent trying to get quantum out of the lab and onto the stage.",[72443,77837,77838],{"avatar":104,"name":75233,"role":75234,"username":73742},[18,77839,75311],{},[13,77841,77843],{"id":77842},"what-makes-a-pattern-quantum","What makes a pattern quantum",[18,77845,77846],{},"Cellular automata evolve a grid of cells by a fixed local rule, and generative artists have long prized them for the intricate, self-organizing patterns they throw off. Partitioned quantum cellular automata run that idea on a quantum circuit: cells are qubits, the update rule is a small circuit applied across overlapping partitions, and the state evolves through superposition and interference instead of a classical lookup table.",[18,77848,77849],{},"The result is material that is coherent without being predictable. The patterns carry real quantum structure, correlations produced by entanglement and shaped by phase interference, that a classical random-number generator cannot reproduce. Three gates do most of the work:",[74616,77851,77853],{"lead":77852},"A PQCA update rule is built from a handful of gates, applied cell by cell.",[41852,77854,77855,77863],{},[41855,77856,77857],{},[41858,77858,77859,77861],{},[41861,77860,74627],{},[41861,77862,74630],{},[41868,77864,77865,77873,77881],{},[41858,77866,77867,77870],{},[41873,77868,77869],{},"Hadamard",[41873,77871,77872],{},"Superposition, spreads each cell across its possible states.",[41858,77874,77875,77878],{},[41873,77876,77877],{},"CNOT \u002F CX",[41873,77879,77880],{},"Entanglement, couples neighbouring cells across partition borders.",[41858,77882,77883,77886],{},[41873,77884,77885],{},"RZ \u002F phase",[41873,77887,77888],{},"Interference, shapes how amplitudes reinforce and cancel.",[72443,77890,77891],{"avatar":104,"name":75233,"role":75234,"username":73742},[18,77892,77893],{},"One of the creative outputs of my thesis was cellular automata. They're really good for demonstrating, because you've got the visual element. Partitioned quantum cellular automata are really effective at some interesting, unfolding, immersive algorithms.",[13,77895,77897],{"id":77896},"a-cellular-automaton-in-a-dozen-lines","A cellular automaton in a dozen lines",[18,77899,77900,77901,77904],{},"The pattern data comes from a Qiskit and ",[1031,77902,77903],{},"pqca"," script small enough to read in one sitting. It lays out a one-dimensional line of qubits, defines the two-qubit circuit applied to every cell, and steps the automaton forward, sampling one state per step. Edit the parameters, run it, and watch the excitation spread.",[831,77906],{"caption":77907,"no":835,"poster":77908,"video":75316},"A PQCA dataset loaded into Satori and scripted live, grid on the right, script on the left. Press play, sound on.","\u002F_content\u002Fimages\u002Fquantum-patterns\u002Fdemo-poster.webp",[493,77910,77913],{"name":77911,"run-href":77912,"tag":72511},"pqca_quickstart.py","https:\u002F\u002Fgithub.com\u002FItaborala\u002Fsatori-pqca-generator",[498,77914,77916],{"className":500,"code":77915,"language":502,"meta":72515,"style":104},"# PQCA quickstart: a partitioned quantum cellular automaton in a dozen lines.\nimport qiskit\nimport pqca\n\n# Parameters\nNUM_QUBITS = 10          # a 1-D line of qubits\nCELL_SIZE  = 2           # qubits per cell\nSTEPS      = 6\nINITIAL = [1] + [0] * (NUM_QUBITS - 1)   # one excitation at the left edge\n\n# The circuit applied to every cell: a single CX on 2 qubits.\ncell = qiskit.QuantumCircuit(CELL_SIZE)\ncell.cx(0, 1)\n\n# Two offset tessellations couple the update across cell borders.\ntes = pqca.tessellation.one_dimensional(NUM_QUBITS, CELL_SIZE)\nframes = [\n    pqca.UpdateFrame(tes, cell),\n    pqca.UpdateFrame(tes.shifted_by(1), cell),\n]\n\n# Unitary mode evolves the statevector locally, sampling one state per step.\nautomaton = pqca.Automaton(INITIAL, frames, mode=pqca.UnitaryPQCA())\n\nprint(f\"t=0  {INITIAL}\")\nfor t in range(1, STEPS + 1):\n    print(f\"t={t}  {next(automaton)}\")\n",[504,77917,77918,77923,77930,77937,77941,77946,77958,77970,77980,78016,78020,78025,78044,78068,78072,78077,78100,78109,78120,78147,78151,78155,78160,78201,78205,78223,78248],{"__ignoreMap":104},[507,77919,77920],{"class":509,"line":510},[507,77921,77922],{"class":562},"# PQCA quickstart: a partitioned quantum cellular automaton in a dozen lines.\n",[507,77924,77925,77927],{"class":509,"line":105},[507,77926,514],{"class":513},[507,77928,77929],{"class":517}," qiskit\n",[507,77931,77932,77934],{"class":509,"line":540},[507,77933,514],{"class":513},[507,77935,77936],{"class":517}," pqca\n",[507,77938,77939],{"class":509,"line":553},[507,77940,556],{"emptyLinePlaceholder":133},[507,77942,77943],{"class":509,"line":559},[507,77944,77945],{"class":562},"# Parameters\n",[507,77947,77948,77950,77952,77955],{"class":509,"line":566},[507,77949,73243],{"class":583},[507,77951,1423],{"class":572},[507,77953,77954],{"class":583}," 10",[507,77956,77957],{"class":562},"          # a 1-D line of qubits\n",[507,77959,77960,77963,77965,77967],{"class":509,"line":590},[507,77961,77962],{"class":583},"CELL_SIZE",[507,77964,77138],{"class":572},[507,77966,2316],{"class":583},[507,77968,77969],{"class":562},"           # qubits per cell\n",[507,77971,77972,77975,77978],{"class":509,"line":610},[507,77973,77974],{"class":583},"STEPS",[507,77976,77977],{"class":572},"      =",[507,77979,9014],{"class":583},[507,77981,77982,77985,77987,77989,77991,77993,77995,77997,77999,78001,78003,78005,78007,78009,78011,78013],{"class":509,"line":634},[507,77983,77984],{"class":583},"INITIAL",[507,77986,1423],{"class":572},[507,77988,8427],{"class":517},[507,77990,625],{"class":583},[507,77992,8206],{"class":517},[507,77994,2107],{"class":572},[507,77996,8427],{"class":517},[507,77998,601],{"class":583},[507,78000,8206],{"class":517},[507,78002,2391],{"class":572},[507,78004,58644],{"class":517},[507,78006,73243],{"class":583},[507,78008,65914],{"class":572},[507,78010,1426],{"class":583},[507,78012,67189],{"class":517},[507,78014,78015],{"class":562},"# one excitation at the left edge\n",[507,78017,78018],{"class":509,"line":661},[507,78019,556],{"emptyLinePlaceholder":133},[507,78021,78022],{"class":509,"line":678},[507,78023,78024],{"class":562},"# The circuit applied to every cell: a single CX on 2 qubits.\n",[507,78026,78027,78030,78032,78035,78038,78040,78042],{"class":509,"line":683},[507,78028,78029],{"class":517},"cell ",[507,78031,573],{"class":572},[507,78033,78034],{"class":517}," qiskit.",[507,78036,78037],{"class":576},"QuantumCircuit",[507,78039,580],{"class":517},[507,78041,77962],{"class":583},[507,78043,587],{"class":517},[507,78045,78046,78049,78051,78053,78055,78057,78059,78061],{"class":509,"line":697},[507,78047,78048],{"class":517},"cell.",[507,78050,615],{"class":576},[507,78052,580],{"class":517},[507,78054,601],{"class":583},[507,78056,622],{"class":517},[507,78058,625],{"class":583},[507,78060,3649],{"class":517},[507,78062,72708,78063],{"class":72706,"tabindex":72707},[507,78064,78065,78067],{"class":72711,"role":72712},[154,78066,73431],{}," The CX links the two qubits in a cell so their states become correlated rather than independent: the quantum core of the update rule.",[507,78069,78070],{"class":509,"line":710},[507,78071,556],{"emptyLinePlaceholder":133},[507,78073,78074],{"class":509,"line":715},[507,78075,78076],{"class":562},"# Two offset tessellations couple the update across cell borders.\n",[507,78078,78079,78082,78084,78087,78090,78092,78094,78096,78098],{"class":509,"line":721},[507,78080,78081],{"class":517},"tes ",[507,78083,573],{"class":572},[507,78085,78086],{"class":517}," pqca.tessellation.",[507,78088,78089],{"class":576},"one_dimensional",[507,78091,580],{"class":517},[507,78093,73243],{"class":583},[507,78095,622],{"class":517},[507,78097,77962],{"class":583},[507,78099,587],{"class":517},[507,78101,78102,78105,78107],{"class":509,"line":736},[507,78103,78104],{"class":517},"frames ",[507,78106,573],{"class":572},[507,78108,2177],{"class":517},[507,78110,78111,78114,78117],{"class":509,"line":748},[507,78112,78113],{"class":517},"    pqca.",[507,78115,78116],{"class":576},"UpdateFrame",[507,78118,78119],{"class":517},"(tes, cell),\n",[507,78121,78122,78124,78126,78129,78132,78134,78136,78139],{"class":509,"line":761},[507,78123,78113],{"class":517},[507,78125,78116],{"class":576},[507,78127,78128],{"class":517},"(tes.",[507,78130,78131],{"class":576},"shifted_by",[507,78133,580],{"class":517},[507,78135,625],{"class":583},[507,78137,78138],{"class":517},"), cell),",[507,78140,72708,78141],{"class":72706,"tabindex":72707},[507,78142,78143,78146],{"class":72711,"role":72712},[154,78144,78145],{},"Offset tiling."," Shifting the second tessellation by one qubit lets information cross cell boundaries, so cells couple instead of evolving in isolation.",[507,78148,78149],{"class":509,"line":775},[507,78150,1794],{"class":517},[507,78152,78153],{"class":509,"line":784},[507,78154,556],{"emptyLinePlaceholder":133},[507,78156,78157],{"class":509,"line":796},[507,78158,78159],{"class":562},"# Unitary mode evolves the statevector locally, sampling one state per step.\n",[507,78161,78162,78165,78167,78170,78173,78175,78177,78180,78182,78184,78187,78190,78193],{"class":509,"line":809},[507,78163,78164],{"class":517},"automaton ",[507,78166,573],{"class":572},[507,78168,78169],{"class":517}," pqca.",[507,78171,78172],{"class":576},"Automaton",[507,78174,580],{"class":517},[507,78176,77984],{"class":583},[507,78178,78179],{"class":517},", frames, ",[507,78181,23938],{"class":2155},[507,78183,573],{"class":572},[507,78185,78186],{"class":517},"pqca.",[507,78188,78189],{"class":576},"UnitaryPQCA",[507,78191,78192],{"class":517},"())",[507,78194,72708,78195],{"class":72706,"tabindex":72707},[507,78196,78197,78200],{"class":72711,"role":72712},[154,78198,78199],{},"Sampled, not collapsed."," Unitary mode evolves the full statevector and samples one outcome per step without collapsing it, so the automaton keeps its quantum structure as it runs. No backend or account needed.",[507,78202,78203],{"class":509,"line":1352},[507,78204,556],{"emptyLinePlaceholder":133},[507,78206,78207,78209,78211,78213,78216,78219,78221],{"class":509,"line":1357},[507,78208,8525],{"class":572},[507,78210,580],{"class":517},[507,78212,22278],{"class":513},[507,78214,78215],{"class":730},"\"t=0  ",[507,78217,78218],{"class":583},"{INITIAL}",[507,78220,22281],{"class":730},[507,78222,587],{"class":517},[507,78224,78225,78227,78230,78232,78234,78236,78238,78240,78242,78244,78246],{"class":509,"line":1362},[507,78226,1630],{"class":513},[507,78228,78229],{"class":517}," t ",[507,78231,1636],{"class":513},[507,78233,8221],{"class":572},[507,78235,580],{"class":517},[507,78237,625],{"class":583},[507,78239,622],{"class":517},[507,78241,77974],{"class":583},[507,78243,8313],{"class":572},[507,78245,1426],{"class":583},[507,78247,1883],{"class":517},[507,78249,78250,78252,78254,78256,78259,78261,78263,78265,78268,78271,78274,78276,78278],{"class":509,"line":1367},[507,78251,2060],{"class":572},[507,78253,580],{"class":517},[507,78255,22278],{"class":513},[507,78257,78258],{"class":730},"\"t=",[507,78260,2810],{"class":583},[507,78262,3298],{"class":517},[507,78264,2872],{"class":583},[507,78266,78267],{"class":583},"  {",[507,78269,78270],{"class":572},"next",[507,78272,78273],{"class":517},"(automaton)",[507,78275,2872],{"class":583},[507,78277,22281],{"class":730},[507,78279,587],{"class":517},[18,78281,78282],{},"Two offset tessellations are the trick: applying the update on one tiling, then a version shifted by a single qubit, lets information cross the cell borders so the automaton actually couples rather than evolving each cell in isolation. Small runs evolve locally as a statevector; larger circuits move onto real IonQ hardware, where the machine's own noise enters the pattern.",[74616,78284,78286],{"lead":78285},"Two open-source, MIT-licensed outputs, built to be run and extended.",[41852,78287,78288,78296],{},[41855,78289,78290],{},[41858,78291,78292,78294],{},[41861,78293,74627],{},[41861,78295,74630],{},[41868,78297,78298,78306,78313,78320],{},[41858,78299,78300,78303],{},[41873,78301,78302],{},"Generator",[41873,78304,78305],{},"Python and Qiskit with the pqca library, PQCA circuits exported as JSON pattern data.",[41858,78307,78308,78310],{},[41873,78309,75155],{},[41873,78311,78312],{},"A fork of Satori, terse JavaScript live coding, synthesized in the browser via Web Audio.",[41858,78314,78315,78317],{},[41873,78316,72293],{},[41873,78318,78319],{},"Local statevector simulator, or IonQ hardware for larger circuits.",[41858,78321,78322,78325],{},[41873,78323,78324],{},"Datasets",[41873,78326,78327],{},"Pre-computed PQCA runs, loaded as a first-class live-coding primitive.",[13,78329,78331],{"id":78330},"from-circuit-to-sound","From circuit to sound",[18,78333,78334],{},"Each run of the automaton produces a stream of measurement outcomes: probabilities, amplitudes, and phase values, step by step. Those numbers are exported as JSON and become the musical material: in the browser fork of Satori, they drive pitch, rhythm, timbre, texture, and space. Satori is a terse JavaScript live-coding language, so a few lines map a dataset onto sound and you hear the result immediately, synthesized in the browser via the Web Audio API.",[18,78336,78337],{},"Because the pattern is quantum, the music inherits its character: structurally coherent, non-repeating, and full of the correlations that entanglement and interference produce. Peter frames the compositions less as fixed pieces than as systems of relationships and probabilities. More than that, they are a way of carrying ideas further than a written paper can.",[72443,78339,78340],{"avatar":104,"name":75233,"role":75234,"username":73742},[18,78341,75237],{},[13,78343,78345],{"id":78344},"an-invitation-not-instructions","An invitation, not instructions",[18,78347,78348],{},"The project ships as a fork of Satori, pre-loaded with PQCA datasets generated on simulators and on real hardware, plus a set of worked example compositions. Example 00 just puts the data on a grid; from 01 onward, the examples show how to map it into music with the Satori language. Open it in a browser, hit Run, and start hacking. The compositions are written to be pulled apart and remixed, credited to Peter and Paulo's performance project, Teom y Puy.",[72443,78350,78351],{"avatar":104,"name":75233,"role":75234,"username":73742},[18,78352,78353],{},"We fork Satori, develop some PQCA datasets, visualize them, and put musical scripts alongside, presented as an invitation for people to start hacking it about and using them as springboards for their own ideas.",[18,78355,78356],{},"The barrier this removes is the one that has kept quantum computer music largely academic: the Python, the accounts, the hardware setup. Here it is just a browser tab and plain JavaScript. Paulo, who has spent years putting quantum algorithms on stage, sees the same opening.",[72443,78358,78359],{"avatar":104,"name":74407,"role":75325,"username":74409},[18,78360,75328],{},[13,78362,73027],{"id":73026},[18,78364,78365],{},"Everything is open source and MIT-licensed: the Qiskit generator that produces the PQCA datasets, and the Satori fork that turns them into music. Open the live environment in a browser, load an example, and compose with quantum-derived patterns, or generate your own datasets and extend the library.",[73026,78367,78371],{"fork-href":78368,"live-href":78369,"title":78370},"\u002Fu\u002Fcephasteom\u002Fquantum-patterns","https:\u002F\u002Fpqca.cephasteom.co.uk\u002F","Load a pattern. Make it music.",[18,78372,78373,78374],{},"Fork Quantum Patterns, open the Satori environment in your browser, and live-code with datasets drawn from quantum cellular automata. ",[154,78375,73040],{},[953,78377,78378],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":78380},[78381,78382,78383,78384,78385],{"id":77842,"depth":105,"text":77843},{"id":77896,"depth":105,"text":77897},{"id":78330,"depth":105,"text":78331},{"id":78344,"depth":105,"text":78345},{"id":73026,"depth":105,"text":73027},[112,969,73744],[78388,78398],{"username":73742,"name":75233,"role":78389,"avatar":104,"bio":78390,"links":78391},"Project lead · live coder & composer","Peter, who performs as Cephas Teom, is an artist and researcher working across live coding, quantum algorithms, and web audio. His 2025 doctorate at Plymouth's ICCMR, Zen & The Art of Praxis, produced Zen and its successor Satori: browser-based live-coding environments for quantum computer music. He has performed at CTM Festival Berlin, Sónar Barcelona, and the Quantum Itineraries Festival in Nicosia, and co-founded the software lab Lunar.",[78392,78394,78396],{"label":73068,"href":78393},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fcephasteom",{"label":76220,"href":78395},"https:\u002F\u002Fcephasteom.co.uk\u002F",{"label":979,"href":78397},"https:\u002F\u002Fgithub.com\u002Fcephasteom\u002FSatori",{"username":74409,"name":74407,"role":78399,"avatar":104,"bio":78400,"links":78401},"Quantum algorithms & hardware","Paulo is an interdisciplinary researcher working where physics meets music technology. A PhD student at the Cyprus Institute (part of the ERA-chair QUEST grant) in collaboration with DESY, he investigates variational quantum algorithms for high-energy physics and tools for sonifying and visualizing quantum computation. On Quantum Patterns he takes the supportive role: IonQ hardware integration and validating the quantum side.",[78402,78403],{"label":76220,"href":76221},{"label":979,"href":76223},{"username":8,"name":73075,"role":73076,"avatar":73077},"Peter Thomas and Paulo Itaboraí turned partitioned quantum cellular automata into live-coded musical material: coherent, non-repeating patterns you can open in a browser and compose with.","Peter Thomas and Paulo Itaboraí built Quantum Patterns: partitioned quantum cellular automata as live-coded music in a browser fork of Satori. A Qollab project.",{"href":78368,"label":75368},{"image":75315,"alt":75314,"liveUrl":78369},{},"\u002Fblog\u002Fquantum-patterns",[],[78413,78414,78415],{"username":73094,"project":73095,"title":73096,"category":73097,"thumb":73098,"to":73099},{"username":1004,"project":1005,"title":1006,"category":1007,"thumb":1008,"to":1009},{"username":75023,"project":76250,"title":74564,"category":75052,"thumb":76251,"to":74563},{"title":78417,"description":78418},"Quantum Creative Project Showcase: Quantum Patterns","Partitioned quantum cellular automata become live-coded musical patterns you can hack in the browser.","blog\u002Fquantum-patterns",[73758,78421,143],"live-coding","rrIfOSVZ2WqPNBmRuoJ7GIvy3bFEeP0x8hTWcE3IiOg",{"id":78424,"title":78425,"authors":78426,"body":78427,"breadcrumb":79053,"builders":79054,"byline":79076,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":79077,"description":79078,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":79079,"hero":79081,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":79084,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":79085,"publishDate":79086,"readingTime":74132,"related":79087,"relatedProjects":79088,"seo":79094,"stem":79097,"tags":79098,"track":116,"trackName":116,"__hash__":79100},"blog\u002Fblog\u002Fqorbital.md","Project Showcase: qOrbital",[74390],{"type":10,"value":78428,"toc":79047},[78429,78432,78435,78439,78443,78454,78457,78463,78468,78472,78475,78478,78483,78486,78488,78491,78975,78978,79030,79032,79035,79044],[18,78430,78431],{},"When a chemist draws a molecule, the electrons are not really dots. They are fuzzy clouds of probability. qOrbital computes those clouds for small molecules on real quantum hardware, then lets you look at them.",[18,78433,78434],{},"Pick a molecule (H₂, HeH⁺, LiH), drag a bond-length slider, and a full quantum-chemistry pipeline runs underneath: classical integrals from PySCF, a qubit Hamiltonian from Qiskit Nature, and a ground state found by VQE on IonQ. From that state qOrbital reconstructs the electron density and renders it. It is built by Aryan Bawa and Arnav Singh, two Dartmouth physics students, and its sharpest idea is what it refuses to hide.",[72443,78436,78437],{"avatar":104,"name":74288,"role":74389,"username":104},[18,78438,74393],{},[13,78440,78442],{"id":78441},"two-ways-to-see-an-electron","Two ways to see an electron",[18,78444,78445,78446,78449,78450,78453],{},"What does an orbital actually look like? Quantum mechanics gives two famous answers, and qOrbital shows both from the same wavefunction. The ",[154,78447,78448],{},"Copenhagen"," view is the textbook one: a probability cloud, rendered as a 3D density isosurface. The ",[154,78451,78452],{},"Bohmian"," view takes that same wavefunction, computes a velocity field from its phase, and integrates particle trajectories through it.",[18,78455,78456],{},"Drop a handful of seed particles into the Bohmian view and they do not fill the orbital so much as draw it. Paths bend around nodes and crowd where the bonding density is highest. You set a molecule and a geometry, and the orbital responds in real time as you move the slider, with a live dashboard showing the VQE optimizer working its way toward the ground state.",[831,78458],{"alt":78459,"caption":78460,"no":78461,"src":78462},"The qOrbital interface: a 3D H₂ orbital isosurface threaded with Bohmian trajectories, with the VQE dashboard alongside","The H₂ orbital as an isosurface with Bohmian trajectories, 28 IonQ runs overlaid into the uncertainty cloud. Demo by Aryan Bawa & Arnav Singh.","qOrbital in motion.","\u002F_content\u002Fimages\u002Fqorbital\u002Fscreenshot.webp",[72443,78464,78465],{"avatar":104,"name":74288,"role":74389,"username":104},[18,78466,78467],{},"Two radically different interpretations, one underlying wavefunction, side by side. The connection becomes something you watch instead of something you are told.",[13,78469,78471],{"id":78470},"noise-you-can-see","Noise you can see",[18,78473,78474],{},"This is the part no other visualizer does. Bohmian trajectories need a high-fidelity wavefunction, not just an energy expectation value, so every hardware run carries the full fingerprint of finite measurement statistics. Overlay many independent IonQ runs and you get a trajectory uncertainty cloud.",[18,78476,78477],{},"Where the physics is robust, the trajectories pile up sharp and clean. Near nodes, where the wavefunction is most fragile, they smear out. The error stops being an abstract bar on a chart and becomes a shape you can read.",[72443,78479,78480],{"avatar":104,"name":74288,"role":74389,"username":104},[18,78481,78482],{},"Instead of a dry error bar, you see exactly where a trapped-ion processor is and isn't sure.",[18,78484,78485],{},"To make that explorable without a hardware queue, real runs on IonQ Aria and Forte are bundled as a gallery of independent results per molecule, so visitors can browse genuine hardware output, run-to-run variation and all, while a clean Aer statevector simulator provides the \"this is what perfect looks like\" baseline.",[13,78487,77017],{"id":77016},[18,78489,78490],{},"The pipeline is a real quantum-chemistry stack, end to end: PySCF builds the classical integrals and a Hartree-Fock reference, Qiskit Nature maps the fermionic problem to qubits, VQE finds the ground state, and the optimized state is turned back into a density grid and a Bohmian velocity field for rendering. The core loop is small enough to run yourself, and on Qollab it runs as written:",[493,78492,78495],{"name":78493,"run-href":78494,"tag":72511},"vqe_quickstart.py","\u002Fu\u002Fbawa27\u002Fqorbital",[498,78496,78498],{"className":500,"code":78497,"language":502,"meta":72515,"style":104},"# qOrbital quickstart: find a molecule's ground state with VQE.\nimport numpy as np\nfrom qiskit.primitives import StatevectorEstimator\nfrom qiskit_algorithms import VQE\nfrom qiskit_algorithms.optimizers import SLSQP\nfrom qiskit_nature.second_q.circuit.library import UCCSD, HartreeFock\nfrom qiskit_nature.second_q.drivers import PySCFDriver\nfrom qiskit_nature.second_q.mappers import JordanWignerMapper\n\nMOLECULE, BOND = \"H2\", 0.735          # also: HeH+, LiH\n\n# 1. Classical chemistry: Hartree-Fock + molecular integrals\nproblem = PySCFDriver(atom=f\"H 0 0 0; H 0 0 {BOND}\", basis=\"sto-3g\").run()\n\n# 2. Fermions to qubits\nmapper = JordanWignerMapper()\nqubit_op = mapper.map(problem.hamiltonian.second_q_op())\n\n# 3. UCCSD trial state, from the Hartree-Fock reference\nhf = HartreeFock(problem.num_spatial_orbitals, problem.num_particles, mapper)\nansatz = UCCSD(problem.num_spatial_orbitals, problem.num_particles, mapper, initial_state=hf)\n\ndef show(step, params, energy, meta):\n    print(f\"  iter {step:>3}   E_elec = {energy:+.6f} Ha\")\n\nvqe = VQE(StatevectorEstimator(), ansatz, SLSQP(maxiter=100),\n          callback=show, initial_point=np.zeros(ansatz.num_parameters))\nresult = vqe.compute_minimum_eigenvalue(qubit_op)\n\ntotal = result.eigenvalue.real + problem.hamiltonian.nuclear_repulsion_energy\nprint(f\"  ground state = {total:+.6f} Ha\")\n",[504,78499,78500,78505,78515,78527,78539,78551,78566,78578,78590,78594,78617,78621,78626,78676,78680,78685,78705,78726,78730,78735,78748,78776,78780,78808,78852,78856,78888,78911,78927,78931,78951],{"__ignoreMap":104},[507,78501,78502],{"class":509,"line":510},[507,78503,78504],{"class":562},"# qOrbital quickstart: find a molecule's ground state with VQE.\n",[507,78506,78507,78509,78511,78513],{"class":509,"line":105},[507,78508,514],{"class":513},[507,78510,518],{"class":517},[507,78512,521],{"class":513},[507,78514,524],{"class":517},[507,78516,78517,78519,78522,78524],{"class":509,"line":540},[507,78518,529],{"class":513},[507,78520,78521],{"class":517}," qiskit.primitives ",[507,78523,514],{"class":513},[507,78525,78526],{"class":517}," StatevectorEstimator\n",[507,78528,78529,78531,78534,78536],{"class":509,"line":553},[507,78530,529],{"class":513},[507,78532,78533],{"class":517}," qiskit_algorithms ",[507,78535,514],{"class":513},[507,78537,78538],{"class":583}," VQE\n",[507,78540,78541,78543,78546,78548],{"class":509,"line":559},[507,78542,529],{"class":513},[507,78544,78545],{"class":517}," qiskit_algorithms.optimizers ",[507,78547,514],{"class":513},[507,78549,78550],{"class":583}," SLSQP\n",[507,78552,78553,78555,78558,78560,78563],{"class":509,"line":566},[507,78554,529],{"class":513},[507,78556,78557],{"class":517}," qiskit_nature.second_q.circuit.library ",[507,78559,514],{"class":513},[507,78561,78562],{"class":583}," UCCSD",[507,78564,78565],{"class":517},", HartreeFock\n",[507,78567,78568,78570,78573,78575],{"class":509,"line":590},[507,78569,529],{"class":513},[507,78571,78572],{"class":517}," qiskit_nature.second_q.drivers ",[507,78574,514],{"class":513},[507,78576,78577],{"class":517}," PySCFDriver\n",[507,78579,78580,78582,78585,78587],{"class":509,"line":610},[507,78581,529],{"class":513},[507,78583,78584],{"class":517}," qiskit_nature.second_q.mappers ",[507,78586,514],{"class":513},[507,78588,78589],{"class":517}," JordanWignerMapper\n",[507,78591,78592],{"class":509,"line":634},[507,78593,556],{"emptyLinePlaceholder":133},[507,78595,78596,78599,78601,78604,78606,78609,78611,78614],{"class":509,"line":661},[507,78597,78598],{"class":583},"MOLECULE",[507,78600,622],{"class":517},[507,78602,78603],{"class":583},"BOND",[507,78605,1423],{"class":572},[507,78607,78608],{"class":730}," \"H2\"",[507,78610,622],{"class":517},[507,78612,78613],{"class":583},"0.735",[507,78615,78616],{"class":562},"          # also: HeH+, LiH\n",[507,78618,78619],{"class":509,"line":678},[507,78620,556],{"emptyLinePlaceholder":133},[507,78622,78623],{"class":509,"line":683},[507,78624,78625],{"class":562},"# 1. Classical chemistry: Hartree-Fock + molecular integrals\n",[507,78627,78628,78631,78633,78636,78638,78641,78643,78645,78648,78651,78653,78655,78657,78659,78662,78664,78666,78668],{"class":509,"line":697},[507,78629,78630],{"class":517},"problem ",[507,78632,573],{"class":572},[507,78634,78635],{"class":576}," PySCFDriver",[507,78637,580],{"class":517},[507,78639,78640],{"class":2155},"atom",[507,78642,573],{"class":572},[507,78644,22278],{"class":513},[507,78646,78647],{"class":730},"\"H 0 0 0; H 0 0 ",[507,78649,78650],{"class":583},"{BOND}",[507,78652,22281],{"class":730},[507,78654,622],{"class":517},[507,78656,59743],{"class":2155},[507,78658,573],{"class":572},[507,78660,78661],{"class":730},"\"sto-3g\"",[507,78663,14176],{"class":517},[507,78665,22501],{"class":576},[507,78667,66172],{"class":517},[507,78669,72708,78670],{"class":72706,"tabindex":72707},[507,78671,78672,78675],{"class":72711,"role":72712},[154,78673,78674],{},"Classical setup."," PySCF runs Hartree-Fock and builds the molecular integrals. This is also the classical baseline qOrbital compares against.",[507,78677,78678],{"class":509,"line":710},[507,78679,556],{"emptyLinePlaceholder":133},[507,78681,78682],{"class":509,"line":715},[507,78683,78684],{"class":562},"# 2. Fermions to qubits\n",[507,78686,78687,78690,78692,78695,78697],{"class":509,"line":721},[507,78688,78689],{"class":517},"mapper ",[507,78691,573],{"class":572},[507,78693,78694],{"class":576}," JordanWignerMapper",[507,78696,66172],{"class":517},[507,78698,72708,78699],{"class":72706,"tabindex":72707},[507,78700,78701,78704],{"class":72711,"role":72712},[154,78702,78703],{},"Mapping."," Turns the fermionic Hamiltonian into qubit operators. LiH instead uses parity mapping with a 2-qubit reduction to keep the qubit count down.",[507,78706,78707,78710,78712,78715,78718,78721,78724],{"class":509,"line":736},[507,78708,78709],{"class":517},"qubit_op ",[507,78711,573],{"class":572},[507,78713,78714],{"class":517}," mapper.",[507,78716,78717],{"class":576},"map",[507,78719,78720],{"class":517},"(problem.hamiltonian.",[507,78722,78723],{"class":576},"second_q_op",[507,78725,22087],{"class":517},[507,78727,78728],{"class":509,"line":748},[507,78729,556],{"emptyLinePlaceholder":133},[507,78731,78732],{"class":509,"line":761},[507,78733,78734],{"class":562},"# 3. UCCSD trial state, from the Hartree-Fock reference\n",[507,78736,78737,78740,78742,78745],{"class":509,"line":775},[507,78738,78739],{"class":517},"hf ",[507,78741,573],{"class":572},[507,78743,78744],{"class":576}," HartreeFock",[507,78746,78747],{"class":517},"(problem.num_spatial_orbitals, problem.num_particles, mapper)\n",[507,78749,78750,78753,78755,78757,78760,78763,78765,78768],{"class":509,"line":784},[507,78751,78752],{"class":517},"ansatz ",[507,78754,573],{"class":572},[507,78756,78562],{"class":576},[507,78758,78759],{"class":517},"(problem.num_spatial_orbitals, problem.num_particles, mapper, ",[507,78761,78762],{"class":2155},"initial_state",[507,78764,573],{"class":572},[507,78766,78767],{"class":517},"hf)",[507,78769,72708,78770],{"class":72706,"tabindex":72707},[507,78771,78772,78775],{"class":72711,"role":72712},[154,78773,78774],{},"Ansatz."," UCCSD is the trial wavefunction. VQE tunes its parameters to push the energy down to the true ground state.",[507,78777,78778],{"class":509,"line":796},[507,78779,556],{"emptyLinePlaceholder":133},[507,78781,78782,78784,78787,78789,78792,78794,78796,78798,78801,78803,78806],{"class":509,"line":809},[507,78783,1370],{"class":513},[507,78785,78786],{"class":576}," show",[507,78788,580],{"class":517},[507,78790,78791],{"class":1382},"step",[507,78793,622],{"class":517},[507,78795,76378],{"class":1382},[507,78797,622],{"class":517},[507,78799,78800],{"class":1382},"energy",[507,78802,622],{"class":517},[507,78804,78805],{"class":1382},"meta",[507,78807,1883],{"class":517},[507,78809,78810,78812,78814,78816,78819,78821,78823,78825,78827,78830,78832,78834,78837,78839,78842,78844],{"class":509,"line":1352},[507,78811,2060],{"class":572},[507,78813,580],{"class":517},[507,78815,22278],{"class":513},[507,78817,78818],{"class":730},"\"  iter ",[507,78820,2810],{"class":583},[507,78822,78791],{"class":517},[507,78824,77671],{"class":513},[507,78826,2872],{"class":583},[507,78828,78829],{"class":730},"   E_elec = ",[507,78831,2810],{"class":583},[507,78833,78800],{"class":517},[507,78835,78836],{"class":513},":+.6f",[507,78838,2872],{"class":583},[507,78840,78841],{"class":730}," Ha\"",[507,78843,3649],{"class":517},[507,78845,72708,78846],{"class":72706,"tabindex":72707},[507,78847,78848,78851],{"class":72711,"role":72712},[154,78849,78850],{},"The descent."," VQE walks downhill: each step the optimizer nudges the circuit and the energy drops. This callback feeds the live convergence dashboard.",[507,78853,78854],{"class":509,"line":1357},[507,78855,556],{"emptyLinePlaceholder":133},[507,78857,78858,78861,78863,78866,78868,78871,78874,78877,78879,78882,78884,78886],{"class":509,"line":1362},[507,78859,78860],{"class":517},"vqe ",[507,78862,573],{"class":572},[507,78864,78865],{"class":576}," VQE",[507,78867,580],{"class":517},[507,78869,78870],{"class":576},"StatevectorEstimator",[507,78872,78873],{"class":517},"(), ansatz, ",[507,78875,78876],{"class":576},"SLSQP",[507,78878,580],{"class":517},[507,78880,78881],{"class":2155},"maxiter",[507,78883,573],{"class":572},[507,78885,5682],{"class":583},[507,78887,14202],{"class":517},[507,78889,78890,78893,78895,78898,78901,78903,78906,78908],{"class":509,"line":1367},[507,78891,78892],{"class":2155},"          callback",[507,78894,573],{"class":572},[507,78896,78897],{"class":517},"show, ",[507,78899,78900],{"class":2155},"initial_point",[507,78902,573],{"class":572},[507,78904,78905],{"class":517},"np.",[507,78907,2149],{"class":576},[507,78909,78910],{"class":517},"(ansatz.num_parameters))\n",[507,78912,78913,78916,78918,78921,78924],{"class":509,"line":1379},[507,78914,78915],{"class":517},"result ",[507,78917,573],{"class":572},[507,78919,78920],{"class":517}," vqe.",[507,78922,78923],{"class":576},"compute_minimum_eigenvalue",[507,78925,78926],{"class":517},"(qubit_op)\n",[507,78928,78929],{"class":509,"line":1389},[507,78930,556],{"emptyLinePlaceholder":133},[507,78932,78933,78936,78938,78941,78943,78946],{"class":509,"line":1397},[507,78934,78935],{"class":517},"total ",[507,78937,573],{"class":572},[507,78939,78940],{"class":517}," result.eigenvalue.real ",[507,78942,2107],{"class":572},[507,78944,78945],{"class":517}," problem.hamiltonian.nuclear_repulsion_energy",[507,78947,72708,78948],{"class":72706,"tabindex":72707},[507,78949,78950],{"class":72711,"role":72712},"Electronic energy plus nuclear repulsion is the molecule's ground-state energy. qOrbital rebuilds the electron density and Bohmian trajectories from this state.",[507,78952,78953,78955,78957,78959,78962,78964,78967,78969,78971,78973],{"class":509,"line":1412},[507,78954,8525],{"class":572},[507,78956,580],{"class":517},[507,78958,22278],{"class":513},[507,78960,78961],{"class":730},"\"  ground state = ",[507,78963,2810],{"class":583},[507,78965,78966],{"class":517},"total",[507,78968,78836],{"class":513},[507,78970,2872],{"class":583},[507,78972,78841],{"class":730},[507,78974,587],{"class":517},[18,78976,78977],{},"From the converged state, qOrbital extracts the one-particle reduced density matrix to rebuild the electron density on a 3D grid, computes the de Broglie-Bohm velocity field v = (ℏ\u002Fm) Im(∇ψ\u002Fψ) and integrates it with adaptive Runge-Kutta for the trajectories, then renders both with Three.js on the web and PyVista in the lab.",[74616,78979,78981],{"lead":78980},"qOrbital is open source and MIT-licensed, built to be forked and rerun.",[41852,78982,78983,78991],{},[41855,78984,78985],{},[41858,78986,78987,78989],{},[41861,78988,74627],{},[41861,78990,74630],{},[41868,78992,78993,79000,79007,79014,79022],{},[41858,78994,78995,78997],{},[41873,78996,72215],{},[41873,78998,78999],{},"Qiskit, Qiskit Nature, Qiskit Aer, VQE with a UCCSD ansatz.",[41858,79001,79002,79004],{},[41873,79003,72393],{},[41873,79005,79006],{},"qiskit-ionq, IonQ Aria and Forte trapped-ion processors.",[41858,79008,79009,79011],{},[41873,79010,77737],{},[41873,79012,79013],{},"PySCF, NumPy, SciPy, integrals, Hartree-Fock, and density.",[41858,79015,79016,79019],{},[41873,79017,79018],{},"Render",[41873,79020,79021],{},"Three.js (web) and PyVista \u002F VTK (Jupyter), isosurfaces and trajectory streamlines.",[41858,79023,79024,79027],{},[41873,79025,79026],{},"Method",[41873,79028,79029],{},"de Broglie-Bohm trajectories (Bohm 1952; Wyatt 2005).",[13,79031,73027],{"id":73026},[18,79033,79034],{},"qOrbital is open and forkable on Qollab, the full package is MIT-licensed on GitHub, and you can try the hosted demo right now. Swap the molecule, change the bond length, seed your own particles, and rerun the VQE on a simulator or real IonQ hardware. Supported molecules today are H₂ (the tutorial baseline), HeH⁺ (polar asymmetry), and LiH (the main showcase).",[73026,79036,79039],{"fork-href":78494,"live-href":79037,"title":79038},"https:\u002F\u002Fqorbital.xyz","See an orbital two ways, noise and all.",[18,79040,79041,79042],{},"Fork qOrbital, run VQE on a real trapped-ion processor, and watch the electron density and its Bohmian trajectories take shape. ",[154,79043,73040],{},[953,79045,79046],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":79048},[79049,79050,79051,79052],{"id":78441,"depth":105,"text":78442},{"id":78470,"depth":105,"text":78471},{"id":77016,"depth":105,"text":77017},{"id":73026,"depth":105,"text":73027},[112,969,74277],[79055,79066],{"username":74390,"name":79056,"role":79057,"avatar":104,"bio":79058,"links":79059},"Aryan Bawa","Quantum computing lead","Aryan studies physics and mathematics (modified with computer science) at Dartmouth College, where he is an undergraduate researcher in the Whitfield Group working on quantum algorithms for circuit approximation via the quantum singular value transform. He wrote qsp-proc, an open-source Python\u002FQiskit package for QSP phase-finding, with coursework spanning quantum information, graduate quantum mechanics, quantum optics, and error correction.",[79060,79062,79064],{"label":73068,"href":79061},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fbawa27",{"label":73059,"href":79063},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fabawa2305\u002F",{"label":979,"href":79065},"https:\u002F\u002Fgithub.com\u002Fbawa27",{"name":79067,"role":79068,"avatar":104,"bio":79069,"links":79070},"Arnav Singh","Software & visualization lead","Arnav studies physics and computer science at Dartmouth and brings the full-stack and rendering side. He has production engineering experience at PlayStation, where he co-led a React analytics platform from MVP to deployment, and research experience applying ML to NASA mission proposals and astronomical data at the South African Astronomical Observatory. On qOrbital he bridges the quantum outputs to interactive 3D in TypeScript, React, and Three.js.",[79071,79073],{"label":73059,"href":79072},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Farnav-singh7",{"label":79074,"href":79075},"Portfolio ↗","https:\u002F\u002Farnavsingh0.github.io",{"username":8,"name":73075,"role":73076,"avatar":73077},"Aryan Bawa and Arnav Singh built a visualizer that runs a real quantum-chemistry calculation and renders a molecule's orbital two ways at once, with the hardware's noise on display instead of hidden.","qOrbital runs VQE on IonQ and renders a molecule's orbital as both a probability cloud and Bohmian trajectories, with hardware noise made visible. A Qollab Spring 2026 project.",{"href":78494,"label":79080},"Fork qOrbital",{"image":79082,"alt":79083,"liveUrl":79037},"\u002F_content\u002Fimages\u002Fqorbital\u002Fhero.webp","qOrbital rendering an H2 molecular orbital as an electron-density cloud with Bohmian trajectories, computed with VQE on IonQ hardware",{},"\u002Fblog\u002Fqorbital","2026-05-06",[],[79089,79090,79091],{"username":997,"project":998,"title":999,"category":73752,"thumb":1001,"to":1002},{"username":73101,"project":73102,"title":73103,"category":73097,"thumb":73104,"to":73105},{"username":74372,"project":79092,"title":74261,"category":73752,"thumb":79093,"to":74365},"quantum-advantage-lab","\u002F_content\u002Fimages\u002Fquantum-advantage-lab\u002Fthumbnail.webp",{"title":79095,"description":79096},"Quantum Creative Project Showcase: qOrbital","A molecular orbital visualizer that runs VQE on IonQ and shows the electron as both a probability cloud and Bohmian trajectories, making hardware noise the exhibit.","blog\u002Fqorbital",[79099,143,1019],"chemistry","DOAV-EfwCovBH11vyXObSm6nQxxuqbbgGyc6zIPCnnI",{"id":79102,"title":73782,"authors":79103,"body":79104,"breadcrumb":79480,"builders":79482,"byline":79483,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":79484,"description":79485,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":116,"lessonCount":116,"meta":79486,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":79487,"publishDate":79488,"readingTime":74132,"related":79489,"relatedProjects":116,"seo":79490,"stem":79493,"tags":79494,"track":116,"trackName":116,"__hash__":79496},"blog\u002Fblog\u002Fspring-2026-cohort.md",[8],{"type":10,"value":79105,"toc":79464},[79106,79109,79113,79121,79124,79127,79130,79133,79137,79148,79151,79154,79157,79160,79168,79172,79179,79182,79185,79188,79191,79194,79198,79205,79208,79211,79214,79217,79220,79224,79237,79240,79243,79246,79249,79252,79256,79264,79267,79270,79273,79276,79280,79292,79295,79298,79301,79304,79308,79320,79323,79326,79329,79332,79335,79339,79346,79349,79352,79355,79358,79362,79370,79373,79376,79379,79382,79386,79401,79404,79407,79410,79414,79423,79426,79429,79432,79436,79440,79443,79446,79450,79453,79456],[18,79107,79108],{},"The Spring 2026 cohort is building on real IonQ hardware this season: tools, music, visualizations, and games. Here is the lineup, project by project.",[13,79110,79112],{"id":79111},"qatalyst-game-race-a-quantum-computer-at-route-planning","Qatalyst Game: race a quantum computer at route planning",[18,79114,79115],{},[1031,79116,79117,79120],{},[49,79118,74256],{"href":79119},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fsiti-fariya\u002F"," · Qatalyst Quantum",[18,79122,79123],{},"Browser game racing you against a classical solver and live IonQ hardware on vehicle routing. Five stops feel easy, twenty feel brutal, and that gap between intuition and quantum compute lands viscerally.",[18,79125,79126],{},"The game runs in the browser and sends real workloads to IonQ in the background. Built on React and Leaflet with a QAOA reformulation of the vehicle routing problem.",[18,79128,79129],{},"Siti is a postdoctoral researcher at Heriot-Watt and the founder of Qatalyst Quantum. Her startup has cohorted through Conception X, Microsoft Founders Hub, Quantinuum Q-NET, and the Kipu Quantum Hub.",[18,79131,79132],{},"She spent two years at the Port of Dover building traffic models, which grounded this project's framing.",[13,79134,79136],{"id":79135},"qorbital-see-electrons-as-both-clouds-and-trajectories","qOrbital: see electrons as both clouds and trajectories",[18,79138,79139],{},[1031,79140,79141,79144,79145],{},[49,79142,79056],{"href":79143},"https:\u002F\u002Flinkedin.com\u002Fin\u002Fabawa2305\u002F"," · Dartmouth, with ",[49,79146,79067],{"href":79147},"https:\u002F\u002Flinkedin.com\u002Fin\u002Farnav-singh7",[18,79149,79150],{},"Interactive molecular orbital visualizer powered by VQE runs on IonQ, with both probability-cloud and Bohmian trajectory views.",[18,79152,79153],{},"When a chemist draws a molecule, the electrons are not really dots. They are fuzzy clouds of probability. qOrbital runs real quantum hardware to compute those clouds for simple molecules.",[18,79155,79156],{},"The dual view sets it apart. The Copenhagen interpretation gives you the cloud, the Bohmian gives you the dot moving through it. Both come from the same hardware-derived wavefunction.",[18,79158,79159],{},"Aryan is a Physics and Mathematics student at Dartmouth, working in the Whitfield Group on quantum algorithms for circuit approximation. He is the developer of qsp-proc, an open-source Qiskit package for QSP phase-finding.",[18,79161,79162,79163,79167],{},"The pipeline runs VQE with a UCCSD ansatz and realistic shot budgets, with a clean fallback to simulator. Code lives at ",[49,79164,79166],{"href":79165},"https:\u002F\u002Fgithub.com\u002Fqorbital-lab\u002Fqorbital","github.com\u002Fqorbital-lab\u002Fqorbital",", with hooks for community-contributed molecules.",[13,79169,79171],{"id":79170},"superposition-sequencer-quantum-circuits-as-a-playable-instrument","Superposition Sequencer: quantum circuits as a playable instrument",[18,79173,79174],{},[1031,79175,79176,79178],{},[49,79177,73132],{"href":73728}," · Incomputable",[18,79180,79181],{},"Web-based music sequencer where the quantum circuit is the instrument, including MIDI control and live circuit projection.",[18,79183,79184],{},"Instead of an algorithm with random numbers, it is a real quantum computation producing pattern after pattern. The framing treats quantum hardware as an instrument rather than a science experiment, the line that separates this from gimmick. Building circuits that produce specific sonic textures becomes the creative loop, like patching a synth for specific tones.",[18,79186,79187],{},"Francisco is a mechatronics engineer and creative technologist with fifteen-plus years of experience. His work spans robotics, industrial automation, data products, and interactive installations.",[18,79189,79190],{},"Through his practice Incomputable in Barcelona, he ships tools and prototypes for artists, designers, and research labs. Recent work includes six installations at Sikka Art Festival in Dubai and an EU Starts and MUSAE Residency prototype.",[18,79192,79193],{},"The Sequencer ships with a downloadable sequence library so people without hardware access can still play the patterns.",[13,79195,79197],{"id":79196},"quantum-butterfly-field-the-no-butterfly-effect-made-tangible","Quantum Butterfly Field: the no-butterfly effect, made tangible",[18,79199,79200],{},[1031,79201,79202],{},[49,79203,73880],{"href":79204},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fxinyiz\u002F",[18,79206,79207],{},"Interactive artwork driven by real scrambling circuits, making the no-butterfly effect tangible.",[18,79209,79210],{},"It is a real quantum-information result where small perturbations in a scrambled system can be fully recovered. The aim is to turn abstract quantum information theory into something you can see and sense with your body. The piece sits at the art-science intersection rather than tooling for developers, which broadens the reach.",[18,79212,79213],{},"Xinyi is a multidisciplinary artist and technologist exploring the intersections of nature, spirituality, and computational media. She holds computer science degrees from MIT and the University of British Columbia.",[18,79215,79216],{},"Her engineering work spans Disney, Pixar, and Google. Her research on Generative AI for Computer Animation won Best Paper at SIGGRAPH MIG.",[18,79218,79219],{},"The piece doubles as a teaching artifact for anyone working with quantum scrambling.",[13,79221,79223],{"id":79222},"graft-see-what-fault-tolerant-compilation-actually-looks-like","GraFT: see what fault-tolerant compilation actually looks like",[18,79225,79226],{},[1031,79227,79228,79232,79233],{},[49,79229,79231],{"href":79230},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fdavid-nizovsky\u002F","David Nizovsky"," · Vanderbilt \u002F Superquantum, with ",[49,79234,79236],{"href":79235},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fdaniil-shatokhin-9a69b7297\u002F","Daniil Shatokhin",[18,79238,79239],{},"3D explorable graphs of how a single logical gate expands into thousands of fault-tolerant physical operations.",[18,79241,79242],{},"GraFT renders the full graph including magic state distillation and syndrome extraction. The piece makes the enormous overhead of fault-tolerant quantum computing feel real rather than abstract. That is rare in a notoriously theoretical subfield.",[18,79244,79245],{},"David is a Quantum Applications Engineer at Superquantum, where he leads rmsynth, an open-source library for fault-tolerant circuit synthesis. He is finishing a B.E. in Electrical and Computer Engineering at Vanderbilt. Previously he won the QRISE Challenge for a neutral-atom compilation visualizer.",[18,79247,79248],{},"Daniil is a Vanderbilt student specializing in low-level systems programming and scalable computational architecture. He previously co-engineered the backend for Qubitcoin.",[18,79250,79251],{},"The team has the rmsynth precedent to ship cleanly here.",[13,79253,79255],{"id":79254},"quantumregimeradar-empirical-quantum-kernels-for-market-regimes","QuantumRegimeRadar: empirical quantum kernels for market regimes",[18,79257,79258],{},[1031,79259,79260,79263],{},[49,79261,74989],{"href":79262},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Farkei\u002F",", PhD · Nelnet, with Aaron Ben-Shalom",[18,79265,79266],{},"Quantum kernel-based market regime detection, empirically backtested against GARCH baselines.",[18,79268,79269],{},"The work tries to detect different kinds of market stress, including crashes, recoveries, and sideways grind, using real historical data. It is grounded in empirical backtesting rather than the usual quantum-finance hype. The aim is to show where quantum kernels help and where they do not.",[18,79271,79272],{},"Alireza holds a Ph.D. in Computer Engineering from the University of Nebraska-Lincoln plus an MBA in Finance. His doctoral work developed a quantum-enhanced framework for GARCH parameter estimation using quantum annealing.",[18,79274,79275],{},"That empirical foundation directly informs the regime library here. Others can plug in their own regimes and rerun backtests on real hardware.",[13,79277,79279],{"id":79278},"musiq-voices-that-harmonize-through-entanglement","Musiq: voices that harmonize through entanglement",[18,79281,79282],{},[1031,79283,79284,79287,79288],{},[49,79285,75181],{"href":79286},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Ftomoya-hatanaka\u002F",", with ",[49,79289,79291],{"href":79290},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Femmanuellaadams\u002F","Emmanuella Adams",[18,79293,79294],{},"Polyphonic generative music using quantum walks for melody and entanglement across voices for harmony, on 30 to 40 qubit circuits.",[18,79296,79297],{},"The voices harmonize in ways no classical computer can produce. That gives you something genuinely quantum to listen to, not just read about. The 30 to 40 qubit circuits genuinely need IonQ hardware. Ballistic spread from quantum walks produces musical leaps impossible classically.",[18,79299,79300],{},"Tomoya recently completed a master's at the University of Tokyo, with research in quantum error correction and hash functions. He also initiated KetQat, an open-source project aimed at making quantum computing more accessible.",[18,79302,79303],{},"Musiq is the strongest hardware story of the cohort's three music projects. The case for why a simulator falls short here is clear.",[13,79305,79307],{"id":79306},"quantum-patterns-pqca-compositions-for-live-coders","Quantum Patterns: PQCA compositions for live coders",[18,79309,79310],{},[1031,79311,79312,79315,79316],{},[49,79313,75233],{"href":79314},"https:\u002F\u002Fgithub.com\u002Fcephasteom"," · ICCMR, with ",[49,79317,79319],{"href":79318},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fpaulo-vitor-itaborai-de-barros-2b45b4105\u002F","Paulo Vitor Itaboraí",[18,79321,79322],{},"Partitioned quantum cellular automata as the engine for live-coded musical composition.",[18,79324,79325],{},"The patterns come from real quantum circuits, not random number generators. The result is music that is structurally coherent but never repeats. The textures are aimed at musicians who want a new instrument, not at physicists.",[18,79327,79328],{},"PQCA outputs are genuinely quantum rather than mapped randomness, which differentiates this from the noise-as-music genre.",[18,79330,79331],{},"Peter is a musician, live coder, and researcher whose doctoral work at the University of Plymouth produced Zen and Satori. Both are web-based live coding environments for quantum computer music. He is affiliated with the Interdisciplinary Centre for Computer Music Research and co-authored research on the Variational Quantum Harmoniser.",[18,79333,79334],{},"The build feeds back into the existing Satori community directly.",[13,79336,79338],{"id":79337},"quantum-market-game-trading-floor-as-an-entanglement-primer","Quantum Market Game: trading floor as an entanglement primer",[18,79340,79341],{},[1031,79342,79343],{},[49,79344,75021],{"href":79345},"https:\u002F\u002Fgithub.com\u002Faadarshv2009",[18,79347,79348],{},"High-school built market simulator that surfaces entanglement and superposition through trading mechanics.",[18,79350,79351],{},"Players make trading decisions while a real quantum computer drives the market behavior. Designed for high school and early college students who know math but have never touched quantum computing.",[18,79353,79354],{},"The familiar market-game format earns its keep, with entanglement-as-correlation framed for classroom students to actually follow. Structured for classroom adoption.",[18,79356,79357],{},"Aadarsh has shipped quantum code solo before. The near-zero compute ask was unusual enough to be worth rewarding on its own.",[13,79359,79361],{"id":79360},"quantumcanvas-a-sandbox-for-the-post-tutorial-now-what","QuantumCanvas: a sandbox for the post-tutorial 'now what'",[18,79363,79364],{},[1031,79365,79366,79369],{},[49,79367,74240],{"href":79368},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fshivanimayekar\u002F"," · Georgia Tech",[18,79371,79372],{},"Drag-and-drop visual sandbox for composing and running new quantum algorithms on real hardware.",[18,79374,79375],{},"Designed for people who have finished quantum tutorials but find actually building new circuits intimidating and repetitive. The post-tutorial gap is real, and the modular-primitives approach gives a way through. It does not require restarting from scratch every time.",[18,79377,79378],{},"Shivani is an M.S. Computer Science student at Georgia Tech. Her quantum work includes winning the QRISE 2024 Infleqtion Challenge and participating in Womanium Quantum AI research.",[18,79380,79381],{},"She co-founded Qtangled and has organized quantum computing workshops for over 250 participants.",[13,79383,79385],{"id":79384},"qcflows-how-measurement-basis-changes-what-you-see","QCFlows: how measurement basis changes what you see",[18,79387,79388],{},[1031,79389,79390,79392,79393,10799,79397],{},[49,79391,79319],{"href":79318}," · Cyprus Institute \u002F DESY, with ",[49,79394,79396],{"href":79395},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fiosifinaangelidi\u002F","Iosifina Angelidi",[49,79398,79400],{"href":79399},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fkostas-blekos-0a83575\u002F","Kostas Blekos",[18,79402,79403],{},"Interactive visualizer for quantum correlations and basis-dependent measurement views of circuits.",[18,79405,79406],{},"Aimed at researchers and advanced students who want intuition for how entanglement manifests differently under different views. Most quantum visualizers fix one view; this one moves between them, which is the distinct angle.",[18,79408,79409],{},"The team has an existing working prototype, plus DESY and Cyprus Institute affiliations behind it. Paulo also works on Quantum Patterns this season, which gives the cohort a rare cross-project link. We are excited to see how it lands once teams start sharing demos.",[13,79411,79413],{"id":79412},"entangled-body-a-quantum-inspired-body-where-touch-ripples-non-locally","Entangled Body: a quantum-inspired body where touch ripples non-locally",[18,79415,79416],{},[1031,79417,79418,79420,79421],{},[49,79419,72459],{"href":73060}," · HKU, with ",[49,79422,72445],{"href":73071},[18,79424,79425],{},"A point-cloud visualization of a human body where touching one area triggers non-local responses elsewhere, inspired by entanglement.",[18,79427,79428],{},"The user moves the viewpoint and the body reveals itself differently each time. The framing borrows from artist Julian Voss-Andreae's quantum sculpture practice. The piece sits in art-meets-science territory rather than a developer tool.",[18,79430,79431],{},"The team has a working MVP. They plan to wire in real quantum circuit backing as the build progresses. Quantum-conceptual framing here is honest rather than hyped, which is rarer than it should be in this space.",[13,79433,79435],{"id":79434},"quantum-systemic-oracle-quantum-compute-as-an-on-chain-primitive","Quantum Systemic Oracle: quantum compute as an on-chain primitive",[18,79437,79438],{},[1031,79439,74912],{},[18,79441,79442],{},"A blockchain oracle that publishes a quantum-computed 'systemic risk score' for financial markets every day. Smart contracts and prediction markets can use the quantum-derived risk data as a primitive.",[18,79444,79445],{},"The framing is commoditizing quantum compute as an on-chain data feed. Solo developer spanning quantum, smart contracts, oracles, and finance APIs is a high-risk stack. LLM assistance carries parts of the build forward.",[13,79447,79449],{"id":79448},"more-to-come","More to come",[18,79451,79452],{},"More teams are still finalizing paperwork and will join the lineup over the coming weeks. Individual project stories will land on qollab.xyz across the next few weeks and months as teams share progress.",[18,79454,79455],{},"The community Slack is where teams share early code, sketches, and questions. The Fall RFP cycle opens later this year, shifting toward applied domains in optimization and logistics.",[79457,79458],"more-strip",{"all-href":79459,"all-label":79460,"card-sub":79461,"card-title":79462,"href":73797,"title":79463},"\u002Fexplore\u002Fnews","All news & open calls","See the Creative Challenge","Want in on the next one?","More on Qollab",{"title":104,"searchDepth":105,"depth":105,"links":79465},[79466,79467,79468,79469,79470,79471,79472,79473,79474,79475,79476,79477,79478,79479],{"id":79111,"depth":105,"text":79112},{"id":79135,"depth":105,"text":79136},{"id":79170,"depth":105,"text":79171},{"id":79196,"depth":105,"text":79197},{"id":79222,"depth":105,"text":79223},{"id":79254,"depth":105,"text":79255},{"id":79278,"depth":105,"text":79279},{"id":79306,"depth":105,"text":79307},{"id":79337,"depth":105,"text":79338},{"id":79360,"depth":105,"text":79361},{"id":79384,"depth":105,"text":79385},{"id":79412,"depth":105,"text":79413},{"id":79434,"depth":105,"text":79435},{"id":79448,"depth":105,"text":79449},[112,969,79481],"Cohort",[],{"username":8,"name":73075,"role":73076,"avatar":73077},"Quantum projects running on real IonQ hardware this spring: tools, music, visualizations, and games. Here is the lineup, project by project.","Quantum projects running on real IonQ hardware this spring: tools, music, visualizations, and games. Meet the 13 teams in the Qollab Spring 2026 cohort.",{},"\u002Fblog\u002Fspring-2026-cohort","2026-05-05",[],{"title":79491,"description":79492},"Meet the Qollab Spring 2026 cohort","13 teams building on real IonQ hardware: quantum tools, music, visualizations, and games. The Spring 2026 Creative Challenge cohort, project by project.","blog\u002Fspring-2026-cohort",[79495,142,143],"cohort","1bMbFrB-bv6rguDRhLbfhQ7q2TzNzeQO_uhBbhnSyQ0",{"id":79498,"title":79499,"authors":79500,"body":79501,"breadcrumb":80144,"builders":80145,"byline":80153,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":80154,"description":80155,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":80156,"hero":80158,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":80159,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":80160,"publishDate":80161,"readingTime":993,"related":80162,"relatedProjects":80163,"seo":80169,"stem":80172,"tags":80173,"track":116,"trackName":116,"__hash__":80175},"blog\u002Fblog\u002Fquantum-market-game.md","Project Showcase: Quantum Market Game",[75023],{"type":10,"value":79502,"toc":80138},[79503,79506,79509,79514,79518,79521,79524,79529,79532,79536,79539,79542,80057,80102,80105,80107,80110,80115,80118,80120,80123,80127,80136],[18,79504,79505],{},"The Quantum Market Game takes the classic prisoner's dilemma and runs it on a quantum computer. Two traders each decide to buy or sell, but here they are qubits, held in a superposition of both at once.",[18,79507,79508],{},"Turn on entanglement and their choices become linked, so the game settles into outcomes that simply do not exist in classical game theory. It is built for students who know some math but have never touched quantum computing, and it is the work of Aadarsh Venkat Ramanan, a rising high-school senior in Frisco, Texas, who found his way into quantum through an unusually good introduction.",[72443,79510,79511],{"avatar":75020,"name":75021,"role":75022,"username":75023},[18,79512,79513],{},"I got into quantum through a meeting with Marco Pistoia, who worked with my mom at JP Morgan Chase. He taught me the basics, I became really interested, and I did summer research at the University of Texas at Dallas.",[13,79515,79517],{"id":79516},"a-market-in-superposition","A market in superposition",[18,79519,79520],{},"In the classical prisoner's dilemma, two players each make one discrete choice and a payoff table decides who wins. Aadarsh kept that skeleton but swapped the players for traders and the choices for quantum states. Each trader is a qubit, and instead of committing to buy or sell, they sit in a superposition of both, weighted by a probability the player controls.",[18,79522,79523],{},"You drive the game by setting those probabilities, then watching what the market does. Crucially, you can also flip entanglement on and off, and that switch is the whole point: with it on, one trader's leaning toward buying pulls on the other's outcome, producing correlations no pair of independent classical players could show.",[72443,79525,79526],{"avatar":75020,"name":75021,"role":75022,"username":75023},[18,79527,79528],{},"The novelty comes from using quantum-mechanics principles in a classical prisoner's dilemma. It lets the players reach final outcomes that are not possible in classical game theory.",[18,79530,79531],{},"That makes the two hardest ideas in quantum computing something you can feel by playing. Superposition is the trader who has not decided yet; entanglement is the toggle that ties two traders' fates together.",[13,79533,79535],{"id":79534},"two-qubits-two-traders","Two qubits, two traders",[18,79537,79538],{},"The circuit underneath is small enough to read in one sitting, which is the point. Each trader gets one rotation that sets their buy\u002Fsell mix, an optional entangling gate links them, and a measurement opens the market. On Qollab it runs on a simulator or real IonQ hardware exactly as written:",[831,79540],{"caption":79541,"no":835,"poster":75015,"video":75016},"Running the game and reading the outcome table: each trader's buy\u002Fsell probability and who came out ahead. Press play, sound on.",[493,79543,79546],{"name":79544,"run-href":79545,"tag":496},"simulator.py","\u002Fu\u002Fq-aad\u002Fquantum-market-game-theory-sim",[498,79547,79549],{"className":500,"code":79548,"language":502,"meta":104,"style":104},"# 'backend' is pre-created from the \"Select QPU\" dropdown below.\nfrom qiskit import QuantumCircuit\nfrom qiskit.providers.jobstatus import JobStatus\nimport numpy as np, time\n\n# Payoff for (trader 1, trader 2) by outcome — 0 = sell, 1 = buy.\nPAYOFF = {\"00\": (-1, -1), \"01\": (3, 1), \"10\": (1, 3), \"11\": (2, 2)}\n\ndef market(angle_1, angle_2, entangle):\n    qc = QuantumCircuit(2, 2)\n    qc.ry(angle_1, 0)\n    qc.ry(angle_2, 1)\n    if entangle:\n        qc.cx(0, 1)\n    qc.measure([0, 1], [0, 1])\n    return qc\n\nqc = market(np.pi\u002F2, np.pi\u002F2, entangle=True)   # both undecided, entangled\njob = backend.run(qc, shots=1000)\nwhile job.status() is not JobStatus.DONE:\n    time.sleep(2)\ncounts = job.result().get_counts()\n\n# Expected payoff across every possible market\nep1 = ep2 = 0.0\nfor bits, c in counts.items():\n    p1, p2 = PAYOFF[bits]\n    ep1 += (c \u002F 1000) * p1; ep2 += (c \u002F 1000) * p2\nprint(f\"Expected payoff — trader 1: {ep1:.2f}, trader 2: {ep2:.2f}\")\n",[504,79550,79551,79556,79566,79576,79587,79591,79596,79663,79667,79691,79709,79729,79742,79749,79772,79806,79812,79816,79851,79878,79899,79912,79929,79933,79938,79952,79966,79979,80021],{"__ignoreMap":104},[507,79552,79553],{"class":509,"line":510},[507,79554,79555],{"class":562},"# 'backend' is pre-created from the \"Select QPU\" dropdown below.\n",[507,79557,79558,79560,79562,79564],{"class":509,"line":105},[507,79559,529],{"class":513},[507,79561,532],{"class":517},[507,79563,514],{"class":513},[507,79565,537],{"class":517},[507,79567,79568,79570,79572,79574],{"class":509,"line":540},[507,79569,529],{"class":513},[507,79571,73222],{"class":517},[507,79573,514],{"class":513},[507,79575,73227],{"class":517},[507,79577,79578,79580,79582,79584],{"class":509,"line":553},[507,79579,514],{"class":513},[507,79581,518],{"class":517},[507,79583,521],{"class":513},[507,79585,79586],{"class":517}," np, time\n",[507,79588,79589],{"class":509,"line":559},[507,79590,556],{"emptyLinePlaceholder":133},[507,79592,79593],{"class":509,"line":566},[507,79594,79595],{"class":562},"# Payoff for (trader 1, trader 2) by outcome — 0 = sell, 1 = buy.\n",[507,79597,79598,79601,79603,79605,79608,79611,79613,79615,79617,79619,79621,79623,79626,79628,79630,79632,79634,79636,79639,79641,79643,79645,79647,79649,79652,79654,79656,79658,79660],{"class":509,"line":590},[507,79599,79600],{"class":583},"PAYOFF",[507,79602,1423],{"class":572},[507,79604,73290],{"class":517},[507,79606,79607],{"class":730},"\"00\"",[507,79609,79610],{"class":517},": (",[507,79612,2367],{"class":572},[507,79614,625],{"class":583},[507,79616,622],{"class":517},[507,79618,2367],{"class":572},[507,79620,625],{"class":583},[507,79622,2213],{"class":517},[507,79624,79625],{"class":730},"\"01\"",[507,79627,79610],{"class":517},[507,79629,8226],{"class":583},[507,79631,622],{"class":517},[507,79633,625],{"class":583},[507,79635,2213],{"class":517},[507,79637,79638],{"class":730},"\"10\"",[507,79640,79610],{"class":517},[507,79642,625],{"class":583},[507,79644,622],{"class":517},[507,79646,8226],{"class":583},[507,79648,2213],{"class":517},[507,79650,79651],{"class":730},"\"11\"",[507,79653,79610],{"class":517},[507,79655,584],{"class":583},[507,79657,622],{"class":517},[507,79659,584],{"class":583},[507,79661,79662],{"class":517},")}\n",[507,79664,79665],{"class":509,"line":610},[507,79666,556],{"emptyLinePlaceholder":133},[507,79668,79669,79671,79674,79676,79679,79681,79684,79686,79689],{"class":509,"line":634},[507,79670,1370],{"class":513},[507,79672,79673],{"class":576}," market",[507,79675,580],{"class":517},[507,79677,79678],{"class":1382},"angle_1",[507,79680,622],{"class":517},[507,79682,79683],{"class":1382},"angle_2",[507,79685,622],{"class":517},[507,79687,79688],{"class":1382},"entangle",[507,79690,1883],{"class":517},[507,79692,79693,79695,79697,79699,79701,79703,79705,79707],{"class":509,"line":661},[507,79694,72833],{"class":517},[507,79696,573],{"class":572},[507,79698,577],{"class":576},[507,79700,580],{"class":517},[507,79702,584],{"class":583},[507,79704,622],{"class":517},[507,79706,584],{"class":583},[507,79708,587],{"class":517},[507,79710,79711,79713,79715,79718,79720,79722],{"class":509,"line":678},[507,79712,21867],{"class":517},[507,79714,639],{"class":576},[507,79716,79717],{"class":517},"(angle_1, ",[507,79719,601],{"class":583},[507,79721,3649],{"class":517},[507,79723,72708,79724],{"class":72706,"tabindex":72707},[507,79725,79726,79728],{"class":72711,"role":72712},[154,79727,73373],{}," Each trader is one qubit. The RY angle sets their mix of sell (0) and buy (1): 0 is a certain sell, π a certain buy, π\u002F2 a 50-50 market.",[507,79730,79731,79733,79735,79738,79740],{"class":509,"line":683},[507,79732,21867],{"class":517},[507,79734,639],{"class":576},[507,79736,79737],{"class":517},"(angle_2, ",[507,79739,625],{"class":583},[507,79741,587],{"class":517},[507,79743,79744,79746],{"class":509,"line":697},[507,79745,1717],{"class":513},[507,79747,79748],{"class":517}," entangle:\n",[507,79750,79751,79753,79755,79757,79759,79761,79763,79765],{"class":509,"line":710},[507,79752,72955],{"class":517},[507,79754,615],{"class":576},[507,79756,580],{"class":517},[507,79758,601],{"class":583},[507,79760,622],{"class":517},[507,79762,625],{"class":583},[507,79764,3649],{"class":517},[507,79766,72708,79767],{"class":72706,"tabindex":72707},[507,79768,79769,79771],{"class":72711,"role":72712},[154,79770,73431],{}," The optional CNOT links the two traders, so one trader's outcome shifts the other's: the correlations classical game theory can't produce.",[507,79773,79774,79776,79778,79781,79783,79785,79787,79789,79791,79793,79795,79798],{"class":509,"line":715},[507,79775,21867],{"class":517},[507,79777,72822],{"class":576},[507,79779,79780],{"class":517},"([",[507,79782,601],{"class":583},[507,79784,622],{"class":517},[507,79786,625],{"class":583},[507,79788,75921],{"class":517},[507,79790,601],{"class":583},[507,79792,622],{"class":517},[507,79794,625],{"class":583},[507,79796,79797],{"class":517},"])",[507,79799,72708,79800],{"class":72706,"tabindex":72707},[507,79801,79802,79805],{"class":72711,"role":72712},[154,79803,79804],{},"Open the market."," Measurement collapses the superposition to one outcome per shot: 00, 01, 10, or 11.",[507,79807,79808,79810],{"class":509,"line":721},[507,79809,2504],{"class":513},[507,79811,72990],{"class":517},[507,79813,79814],{"class":509,"line":736},[507,79815,556],{"emptyLinePlaceholder":133},[507,79817,79818,79820,79822,79824,79827,79829,79831,79834,79836,79838,79840,79842,79844,79846,79848],{"class":509,"line":748},[507,79819,569],{"class":517},[507,79821,573],{"class":572},[507,79823,79673],{"class":576},[507,79825,79826],{"class":517},"(np.pi",[507,79828,645],{"class":572},[507,79830,584],{"class":583},[507,79832,79833],{"class":517},", np.pi",[507,79835,645],{"class":572},[507,79837,584],{"class":583},[507,79839,622],{"class":517},[507,79841,79688],{"class":2155},[507,79843,573],{"class":572},[507,79845,13878],{"class":583},[507,79847,67189],{"class":517},[507,79849,79850],{"class":562},"# both undecided, entangled\n",[507,79852,79853,79855,79857,79859,79861,79864,79866,79868,79871,79873],{"class":509,"line":761},[507,79854,23964],{"class":517},[507,79856,573],{"class":572},[507,79858,73487],{"class":517},[507,79860,22501],{"class":576},[507,79862,79863],{"class":517},"(qc, ",[507,79865,68762],{"class":2155},[507,79867,573],{"class":572},[507,79869,79870],{"class":583},"1000",[507,79872,3649],{"class":517},[507,79874,72708,79875],{"class":72706,"tabindex":72707},[507,79876,79877],{"class":72711,"role":72712},"Runs on a simulator or real IonQ hardware through Qollab. Aadarsh used IonQ credits to compare the clean simulator against hardware noise.",[507,79879,79880,79883,79885,79887,79889,79891,79893,79895,79897],{"class":509,"line":775},[507,79881,79882],{"class":513},"while",[507,79884,23993],{"class":517},[507,79886,73518],{"class":576},[507,79888,1677],{"class":517},[507,79890,37008],{"class":513},[507,79892,21980],{"class":513},[507,79894,73527],{"class":517},[507,79896,73530],{"class":583},[507,79898,1728],{"class":517},[507,79900,79901,79904,79906,79908,79910],{"class":509,"line":784},[507,79902,79903],{"class":517},"    time.",[507,79905,73540],{"class":576},[507,79907,580],{"class":517},[507,79909,584],{"class":583},[507,79911,587],{"class":517},[507,79913,79914,79917,79919,79921,79923,79925,79927],{"class":509,"line":796},[507,79915,79916],{"class":517},"counts ",[507,79918,573],{"class":572},[507,79920,23993],{"class":517},[507,79922,23996],{"class":576},[507,79924,13983],{"class":517},[507,79926,73558],{"class":576},[507,79928,781],{"class":517},[507,79930,79931],{"class":509,"line":809},[507,79932,556],{"emptyLinePlaceholder":133},[507,79934,79935],{"class":509,"line":1352},[507,79936,79937],{"class":562},"# Expected payoff across every possible market\n",[507,79939,79940,79943,79945,79948,79950],{"class":509,"line":1357},[507,79941,79942],{"class":517},"ep1 ",[507,79944,573],{"class":572},[507,79946,79947],{"class":517}," ep2 ",[507,79949,573],{"class":572},[507,79951,58592],{"class":583},[507,79953,79954,79956,79958,79960,79962,79964],{"class":509,"line":1362},[507,79955,1630],{"class":513},[507,79957,73590],{"class":517},[507,79959,1636],{"class":513},[507,79961,73595],{"class":517},[507,79963,22607],{"class":576},[507,79965,1930],{"class":517},[507,79967,79968,79971,79973,79976],{"class":509,"line":1367},[507,79969,79970],{"class":517},"    p1, p2 ",[507,79972,573],{"class":572},[507,79974,79975],{"class":583}," PAYOFF",[507,79977,79978],{"class":517},"[bits]\n",[507,79980,79981,79984,79986,79989,79991,79994,79996,79998,80001,80003,80005,80007,80009,80011,80013,80016],{"class":509,"line":1379},[507,79982,79983],{"class":517},"    ep1 ",[507,79985,2285],{"class":572},[507,79987,79988],{"class":517}," (c ",[507,79990,645],{"class":572},[507,79992,79993],{"class":583}," 1000",[507,79995,655],{"class":517},[507,79997,2391],{"class":572},[507,79999,80000],{"class":517}," p1; ep2 ",[507,80002,2285],{"class":572},[507,80004,79988],{"class":517},[507,80006,645],{"class":572},[507,80008,79993],{"class":583},[507,80010,655],{"class":517},[507,80012,2391],{"class":572},[507,80014,80015],{"class":517}," p2",[507,80017,72708,80018],{"class":72706,"tabindex":72707},[507,80019,80020],{"class":72711,"role":72712},"A prisoner's-dilemma payoff scores each outcome: both buy (11) is the cooperative square, a lone move pays the most. The quantum game blends these across the whole distribution.",[507,80022,80023,80025,80027,80029,80032,80034,80037,80039,80041,80044,80046,80049,80051,80053,80055],{"class":509,"line":1389},[507,80024,8525],{"class":572},[507,80026,580],{"class":517},[507,80028,22278],{"class":513},[507,80030,80031],{"class":730},"\"Expected payoff — trader 1: ",[507,80033,2810],{"class":583},[507,80035,80036],{"class":517},"ep1",[507,80038,70518],{"class":513},[507,80040,2872],{"class":583},[507,80042,80043],{"class":730},", trader 2: ",[507,80045,2810],{"class":583},[507,80047,80048],{"class":517},"ep2",[507,80050,70518],{"class":513},[507,80052,2872],{"class":583},[507,80054,22281],{"class":730},[507,80056,587],{"class":517},[74616,80058,80060],{"lead":80059},"The Quantum Market Game is open source, built to be forked and extended.",[41852,80061,80062,80070],{},[41855,80063,80064],{},[41858,80065,80066,80068],{},[41861,80067,74627],{},[41861,80069,74630],{},[41868,80071,80072,80080,80087,80094],{},[41858,80073,80074,80077],{},[41873,80075,80076],{},"Frontend",[41873,80078,80079],{},"Streamlit, a browser app where you set the probabilities and toggle entanglement.",[41858,80081,80082,80084],{},[41873,80083,72215],{},[41873,80085,80086],{},"Qiskit, a two-qubit circuit, RY state prep plus an optional CNOT.",[41858,80088,80089,80091],{},[41873,80090,72293],{},[41873,80092,80093],{},"Aer simulator, with IonQ runs to compare against hardware noise.",[41858,80095,80096,80099],{},[41873,80097,80098],{},"Payoffs",[41873,80100,80101],{},"A prisoner's-dilemma matrix scored over the measured distribution.",[18,80103,80104],{},"It is deliberately a teaching object rather than a trading model. In the code, the thing a learner changes is the probability of buying or selling; under the hood, that probability becomes the rotation angle on the qubit, so playing with the game and reading the circuit teach the same idea from two directions.",[13,80106,73675],{"id":73674},[18,80108,80109],{},"The current build is intentionally minimal: two traders, two qubits, one clean payoff table. Aadarsh sees it as a starting point rather than a finished thing.",[72443,80111,80112],{"avatar":75020,"name":75021,"role":75022,"username":75023},[18,80113,80114],{},"Right now the game involves two traders and two qubits. In the future this could involve more complex market situations and more traders, and we could add real assets and information to make it more realistic.",[18,80116,80117],{},"That trajectory is also the argument for why a toy matters. A two-qubit market is small enough to understand completely and structured enough to grow, which makes it a clean baseline for thinking about where quantum methods might eventually touch real financial scenarios, the same territory Aadarsh has been exploring in his own research on option pricing and market-regime detection.",[13,80119,73027],{"id":73026},[18,80121,80122],{},"The game is open and forkable on Qollab, the full code is on GitHub, and you can play it in your browser right now. Change the probabilities, flip entanglement on and off, and watch the payoffs move, then open the circuit and rewrite the rules yourself.",[72443,80124,80125],{"avatar":75020,"name":75021,"role":75022,"username":75023},[18,80126,75026],{},[73026,80128,80131],{"fork-href":79545,"live-href":80129,"title":80130},"https:\u002F\u002Fquantum-game-simulator.streamlit.app\u002F","Put two traders in superposition, then entangle them.",[18,80132,80133,80134],{},"Fork the Quantum Market Game, set the odds, toggle entanglement, and run it on real hardware. ",[154,80135,73040],{},[953,80137,73711],{},{"title":104,"searchDepth":105,"depth":105,"links":80139},[80140,80141,80142,80143],{"id":79516,"depth":105,"text":79517},{"id":79534,"depth":105,"text":79535},{"id":73674,"depth":105,"text":73675},{"id":73026,"depth":105,"text":73027},[112,969,74564],[80146],{"username":75023,"name":75021,"role":80147,"avatar":75020,"bio":80148,"links":80149},"Lead developer & designer","Aadarsh is a rising 12th-grader at Lone Star High School in Frisco, Texas, with a long-running interest in math, physics, and quantitative finance. In summer 2025 he did quantum-computing research under Dr Michael Kolodrubetz at UT Dallas, building a discrete portfolio optimiser with QAOA, then continued independently into quantum amplitude estimation (a Monte Carlo option pricer) and quantum support vector machines for detecting market-stress regimes. He presented his option-pricing work at the 2026 Quantum Economy Conference and is back at UT Dallas researching this summer.",[80150,80152],{"label":73068,"href":80151},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fq-aad",{"label":979,"href":79345},{"username":8,"name":73075,"role":73076,"avatar":73077},"Aadarsh Venkat Ramanan built a quantum twist on the prisoner's dilemma: two traders are two qubits, held in a superposition of buy and sell, and entangled into outcomes classical game theory cannot reach.","Aadarsh Venkat Ramanan built a quantum prisoner's dilemma: two traders as two qubits, in superposition of buy and sell, entangled beyond classical game theory.",{"href":79545,"label":80157},"Fork the game",{"image":75015,"alt":75017,"liveUrl":80129},{},"\u002Fblog\u002Fquantum-market-game","2026-05-04",[],[80164,80165,80166],{"username":74914,"project":76253,"title":74523,"category":75052,"thumb":76254,"to":74522},{"username":74336,"project":77810,"title":74228,"category":73752,"thumb":77811,"to":74329},{"username":74354,"project":80167,"title":74245,"category":73752,"thumb":80168,"to":74347},"quantum-courier","\u002F_content\u002Fimages\u002Fquantum-courier\u002Fthumbnail.webp",{"title":80170,"description":80171},"Quantum Creative Project Showcase: Quantum Market Game","A quantum prisoner's dilemma: two traders as two qubits, in superposition of buy and sell, entangled into outcomes classical game theory can't reach.","blog\u002Fquantum-market-game",[75067,143,80174],"games","y6pMVRPpZ_8NAM1B2FR56jHtAedrP6c71wGcwW59Vbg",{"id":80177,"title":80178,"authors":80179,"body":80180,"breadcrumb":80912,"builders":80913,"byline":80926,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":80927,"description":80928,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":80929,"hero":80930,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":80932,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":80933,"publishDate":80934,"readingTime":80935,"related":80936,"relatedProjects":80937,"seo":80941,"stem":80944,"tags":80945,"track":116,"trackName":116,"__hash__":80946},"blog\u002Fblog\u002Fquantum-courier.md","Project Showcase: Quantum Courier",[74354],{"type":10,"value":80181,"toc":80906},[80182,80185,80188,80191,80197,80201,80204,80259,80264,80267,80272,80276,80283,80825,80871,80875,80878,80882,80885,80887,80890,80895,80904],[18,80183,80184],{},"Quantum Courier is a browser game that turns combinatorial optimisation into a five-stage delivery race. Two robots run the same logistics problem, one classical, one quantum, and you watch which solver wins.",[18,80186,80187],{},"Every stage is a real, published formulation pulled from logistics, from assigning pizzas to cutting graphs. And the game refuses to oversell: classical solvers win some stages, quantum methods win others, and it shows both, with the result and the reason on screen.",[18,80189,80190],{},"It is a Spring 2026 challenge project from Dr Siti Fariya, founder of Qatalyst Quantum and a postdoctoral researcher who spent two years optimising real vehicle routing at the Port of Dover. That industry background is where the whole project comes from.",[72443,80192,80194],{"avatar":80193,"name":74256,"role":74353,"username":74354},"\u002F_content\u002Fimages\u002Fbuilders\u002Fsiti-fariya.webp",[18,80195,80196],{},"My background is classical, doing logistics optimisation at the Port of Dover. I got into quantum through a UK government programme for industry, and I saw the gap between real industry cases and quantum, especially in logistics.",[13,80198,80200],{"id":80199},"five-stages-two-solvers","Five stages, two solvers",[18,80202,80203],{},"Each stage is a different optimisation problem, run by a classical planner using proven heuristics and a quantum-inspired one using QUBO-based search. Each has a clear winner, and the game is upfront about who it is:",[74616,80205,80207],{"lead":80206},"Five published formulations, each with an honest result.",[41852,80208,80209,80217],{},[41855,80210,80211],{},[41858,80212,80213,80215],{},[41861,80214,74627],{},[41861,80216,74630],{},[41868,80218,80219,80227,80235,80243,80251],{},[41858,80220,80221,80224],{},[41873,80222,80223],{},"1 · Pizza assignment",[41873,80225,80226],{},"Linear assignment, classical wins (Hungarian algorithm).",[41858,80228,80229,80232],{},[41873,80230,80231],{},"2 · Single-vehicle routing",[41873,80233,80234],{},"TSP, tied at small N, classical scales better.",[41858,80236,80237,80240],{},[41873,80238,80239],{},"3 · Multi-vehicle, time windows",[41873,80241,80242],{},"VRPTW, classical wins (annealing beats QAOA at 25 customers).",[41858,80244,80245,80248],{},[41873,80246,80247],{},"4 · Graph cutting",[41873,80249,80250],{},"MaxCut, quantum wins (+46.2% on real IonQ Forte).",[41858,80252,80253,80256],{},[41873,80254,80255],{},"5 · Combined planning",[41873,80257,80258],{},"Joint QUBO, depends on instance structure.",[831,80260],{"caption":80261,"no":835,"poster":80262,"video":80263},"Quantum Courier in play: a stage runs, the classical and quantum solvers race the same instance, and the game shows the winner and the margin. Press play.","\u002F_content\u002Fimages\u002Fquantum-courier\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F30f10262-f772-4394-841b-365eac4a0c4a",[18,80265,80266],{},"The competitive framing does the teaching. Five stops feel easy; twenty stops feel brutal, and the difference between solvers stops being a claim in a lecture and becomes something you watch happen.",[72443,80268,80269],{"avatar":80193,"name":74256,"role":74353,"username":74354},[18,80270,80271],{},"I wanted the focus to be on teaching quantum itself, so the optimisation is open, especially the classical part. There are already plenty of open papers on that anyway.",[13,80273,80275],{"id":80274},"the-forte-result","The Forte result",[18,80277,80278,80279,80282],{},"Stage 4 is where quantum wins, and it runs on real hardware. The problem is ",[154,80280,80281],{},"MaxCut"," on a 24-node 3-regular graph: split the nodes into two groups so the number of edges crossing between them is as large as possible. Siti ran QAOA at depth p=1 with 4,096 shots on IonQ Forte, and the cut came back 46.2% above the classical baseline. On Qollab the same circuit runs on a simulator or a real QPU exactly as written:",[493,80284,80287],{"name":80285,"run-href":80286,"tag":496},"quantum_courier_maxcut_forte.py","\u002Fu\u002FSitifar\u002Fquantum-game-pizza-race",[498,80288,80290],{"className":500,"code":80289,"language":502,"meta":104,"style":104},"# Stage 4 — MaxCut on a 24-node 3-regular graph, QAOA p=1.\nfrom qiskit import QuantumCircuit\nfrom qiskit.providers.jobstatus import JobStatus\nimport time\n\nN_NODES, SHOTS = 24, 4096\nEDGES = [(0,1), (0,7), (0,14), (1,2), (1,19), # ... 36 edges, 3 per node]\nGAMMA, BETA = 0.393, 0.785      # pre-tuned on a simulator sweep\n\ndef qaoa_circuit(gamma, beta):\n    qc = QuantumCircuit(N_NODES, N_NODES)\n    qc.h(range(N_NODES))\n    for (u, v) in EDGES:                 # cost layer\n        qc.cx(u, v); qc.rz(2 * gamma, v); qc.cx(u, v)\n    for q in range(N_NODES):\n        qc.rx(2 * beta, q)\n    qc.measure(range(N_NODES), range(N_NODES))\n    return qc\n\ndef cut_value(bitstring):\n    bits = [int(b) for b in bitstring[::-1]]\n    return sum(1 for (u, v) in EDGES if bits[u] != bits[v])\n\ncircuit = qaoa_circuit(GAMMA, BETA)\njob = backend.run(circuit, shots=SHOTS)\nwhile job.status() is not JobStatus.DONE:\n    time.sleep(5)\ncounts = job.result().get_counts()\nbest = max(counts, key=cut_value)        # best split in the shot pool\nprint(f\"Best cut: {cut_value(best)} of {len(EDGES)} edges\")\n",[504,80291,80292,80297,80307,80317,80324,80328,80347,80402,80425,80429,80439,80456,80479,80497,80531,80547,80571,80595,80601,80605,80615,80642,80675,80679,80697,80719,80739,80751,80764,80784],{"__ignoreMap":104},[507,80293,80294],{"class":509,"line":510},[507,80295,80296],{"class":562},"# Stage 4 — MaxCut on a 24-node 3-regular graph, QAOA p=1.\n",[507,80298,80299,80301,80303,80305],{"class":509,"line":105},[507,80300,529],{"class":513},[507,80302,532],{"class":517},[507,80304,514],{"class":513},[507,80306,537],{"class":517},[507,80308,80309,80311,80313,80315],{"class":509,"line":540},[507,80310,529],{"class":513},[507,80312,73222],{"class":517},[507,80314,514],{"class":513},[507,80316,73227],{"class":517},[507,80318,80319,80321],{"class":509,"line":553},[507,80320,514],{"class":513},[507,80322,80323],{"class":517}," time\n",[507,80325,80326],{"class":509,"line":559},[507,80327,556],{"emptyLinePlaceholder":133},[507,80329,80330,80333,80335,80337,80339,80342,80344],{"class":509,"line":566},[507,80331,80332],{"class":583},"N_NODES",[507,80334,622],{"class":517},[507,80336,73263],{"class":583},[507,80338,1423],{"class":572},[507,80340,80341],{"class":583}," 24",[507,80343,622],{"class":517},[507,80345,80346],{"class":583},"4096\n",[507,80348,80349,80352,80354,80357,80359,80361,80363,80366,80368,80370,80372,80374,80376,80378,80380,80382,80384,80386,80388,80390,80392,80394,80397,80399],{"class":509,"line":590},[507,80350,80351],{"class":583},"EDGES",[507,80353,1423],{"class":572},[507,80355,80356],{"class":517}," [(",[507,80358,601],{"class":583},[507,80360,2819],{"class":517},[507,80362,625],{"class":583},[507,80364,80365],{"class":517},"), (",[507,80367,601],{"class":583},[507,80369,2819],{"class":517},[507,80371,58240],{"class":583},[507,80373,80365],{"class":517},[507,80375,601],{"class":583},[507,80377,2819],{"class":517},[507,80379,70038],{"class":583},[507,80381,80365],{"class":517},[507,80383,625],{"class":583},[507,80385,2819],{"class":517},[507,80387,584],{"class":583},[507,80389,80365],{"class":517},[507,80391,625],{"class":583},[507,80393,2819],{"class":517},[507,80395,80396],{"class":583},"19",[507,80398,2213],{"class":517},[507,80400,80401],{"class":562},"# ... 36 edges, 3 per node]\n",[507,80403,80404,80407,80409,80412,80414,80417,80419,80422],{"class":509,"line":610},[507,80405,80406],{"class":583},"GAMMA",[507,80408,622],{"class":517},[507,80410,80411],{"class":583},"BETA",[507,80413,70918],{"class":517},[507,80415,80416],{"class":583},"0.393",[507,80418,622],{"class":517},[507,80420,80421],{"class":583},"0.785",[507,80423,80424],{"class":562},"      # pre-tuned on a simulator sweep\n",[507,80426,80427],{"class":509,"line":634},[507,80428,556],{"emptyLinePlaceholder":133},[507,80430,80431,80433,80436],{"class":509,"line":661},[507,80432,1370],{"class":513},[507,80434,80435],{"class":576}," qaoa_circuit",[507,80437,80438],{"class":517},"(gamma, beta):\n",[507,80440,80441,80444,80446,80448,80450,80452,80454],{"class":509,"line":678},[507,80442,80443],{"class":517},"    qc = ",[507,80445,78037],{"class":576},[507,80447,580],{"class":517},[507,80449,80332],{"class":583},[507,80451,622],{"class":517},[507,80453,80332],{"class":583},[507,80455,587],{"class":517},[507,80457,80458,80460,80462,80464,80466,80468,80470,80472],{"class":509,"line":683},[507,80459,21867],{"class":517},[507,80461,596],{"class":576},[507,80463,580],{"class":517},[507,80465,2204],{"class":572},[507,80467,580],{"class":517},[507,80469,80332],{"class":583},[507,80471,71258],{"class":517},[507,80473,72708,80474],{"class":72706,"tabindex":72707},[507,80475,80476,80478],{"class":72711,"role":72712},[154,80477,73373],{}," A Hadamard on every node starts the circuit in an equal mix of all 2^24 ways to split the graph.",[507,80480,80481,80483,80486,80488,80491,80494],{"class":509,"line":697},[507,80482,1916],{"class":513},[507,80484,80485],{"class":517}," (u, v) ",[507,80487,1636],{"class":513},[507,80489,80490],{"class":583}," EDGES",[507,80492,80493],{"class":517},":                 ",[507,80495,80496],{"class":562},"# cost layer\n",[507,80498,80499,80501,80503,80506,80509,80511,80513,80515,80518,80520,80523],{"class":509,"line":710},[507,80500,72955],{"class":517},[507,80502,615],{"class":576},[507,80504,80505],{"class":517},"(u, v); qc.",[507,80507,80508],{"class":576},"rz",[507,80510,580],{"class":517},[507,80512,584],{"class":583},[507,80514,8229],{"class":572},[507,80516,80517],{"class":517}," gamma, v); qc.",[507,80519,615],{"class":576},[507,80521,80522],{"class":517},"(u, v)",[507,80524,72708,80525],{"class":72706,"tabindex":72707},[507,80526,80527,80530],{"class":72711,"role":72712},[154,80528,80529],{},"Cost."," One ZZ term per edge. It rewards a cut edge: the two endpoints landing on opposite sides of the split.",[507,80532,80533,80535,80537,80539,80541,80543,80545],{"class":509,"line":715},[507,80534,1916],{"class":513},[507,80536,22114],{"class":517},[507,80538,1636],{"class":513},[507,80540,8221],{"class":572},[507,80542,580],{"class":517},[507,80544,80332],{"class":583},[507,80546,1883],{"class":517},[507,80548,80549,80551,80554,80556,80558,80560,80563],{"class":509,"line":721},[507,80550,72955],{"class":517},[507,80552,80553],{"class":576},"rx",[507,80555,580],{"class":517},[507,80557,584],{"class":583},[507,80559,8229],{"class":572},[507,80561,80562],{"class":517}," beta, q)",[507,80564,72708,80565],{"class":72706,"tabindex":72707},[507,80566,80567,80570],{"class":72711,"role":72712},[154,80568,80569],{},"Mixer."," Nudges nodes between the two sides so the optimizer can explore different cuts.",[507,80572,80573,80575,80577,80579,80581,80583,80585,80587,80589,80591,80593],{"class":509,"line":736},[507,80574,21867],{"class":517},[507,80576,72822],{"class":576},[507,80578,580],{"class":517},[507,80580,2204],{"class":572},[507,80582,580],{"class":517},[507,80584,80332],{"class":583},[507,80586,2213],{"class":517},[507,80588,2204],{"class":572},[507,80590,580],{"class":517},[507,80592,80332],{"class":583},[507,80594,22540],{"class":517},[507,80596,80597,80599],{"class":509,"line":748},[507,80598,2504],{"class":513},[507,80600,72990],{"class":517},[507,80602,80603],{"class":509,"line":761},[507,80604,556],{"emptyLinePlaceholder":133},[507,80606,80607,80609,80612],{"class":509,"line":775},[507,80608,1370],{"class":513},[507,80610,80611],{"class":576}," cut_value",[507,80613,80614],{"class":517},"(bitstring):\n",[507,80616,80617,80620,80622,80625,80627,80630,80632,80635,80637,80639],{"class":509,"line":784},[507,80618,80619],{"class":517},"    bits = [",[507,80621,1420],{"class":572},[507,80623,80624],{"class":517},"(b) ",[507,80626,1630],{"class":513},[507,80628,80629],{"class":517}," b ",[507,80631,1636],{"class":513},[507,80633,80634],{"class":517}," bitstring[::",[507,80636,2367],{"class":572},[507,80638,625],{"class":583},[507,80640,80641],{"class":517},"]]\n",[507,80643,80644,80646,80648,80650,80652,80654,80656,80658,80660,80662,80665,80667,80670],{"class":509,"line":796},[507,80645,2504],{"class":513},[507,80647,8815],{"class":572},[507,80649,580],{"class":517},[507,80651,625],{"class":583},[507,80653,8774],{"class":513},[507,80655,80485],{"class":517},[507,80657,1636],{"class":513},[507,80659,80490],{"class":583},[507,80661,66162],{"class":513},[507,80663,80664],{"class":517}," bits[u] ",[507,80666,22619],{"class":572},[507,80668,80669],{"class":517}," bits[v])",[507,80671,72708,80672],{"class":72706,"tabindex":72707},[507,80673,80674],{"class":72711,"role":72712},"Counts how many edges a given split cuts. Maximizing this is the whole game of Stage 4.",[507,80676,80677],{"class":509,"line":809},[507,80678,556],{"emptyLinePlaceholder":133},[507,80680,80681,80684,80687,80689,80691,80693,80695],{"class":509,"line":1352},[507,80682,80683],{"class":517},"circuit = ",[507,80685,80686],{"class":576},"qaoa_circuit",[507,80688,580],{"class":517},[507,80690,80406],{"class":583},[507,80692,622],{"class":517},[507,80694,80411],{"class":583},[507,80696,587],{"class":517},[507,80698,80699,80702,80704,80706,80708,80710,80712,80714],{"class":509,"line":1357},[507,80700,80701],{"class":517},"job = backend.",[507,80703,22501],{"class":576},[507,80705,73492],{"class":517},[507,80707,68762],{"class":2155},[507,80709,573],{"class":572},[507,80711,73263],{"class":583},[507,80713,3649],{"class":517},[507,80715,72708,80716],{"class":72706,"tabindex":72707},[507,80717,80718],{"class":72711,"role":72712},"Submitted to IonQ Forte (trapped-ion). On this 24-node graph the QAOA cut came in 46.2% above the classical baseline.",[507,80720,80721,80723,80725,80727,80729,80731,80733,80735,80737],{"class":509,"line":1362},[507,80722,79882],{"class":513},[507,80724,23993],{"class":517},[507,80726,73518],{"class":576},[507,80728,1677],{"class":517},[507,80730,37008],{"class":513},[507,80732,21980],{"class":513},[507,80734,73527],{"class":517},[507,80736,73530],{"class":583},[507,80738,1728],{"class":517},[507,80740,80741,80743,80745,80747,80749],{"class":509,"line":1367},[507,80742,79903],{"class":517},[507,80744,73540],{"class":576},[507,80746,580],{"class":517},[507,80748,58245],{"class":583},[507,80750,587],{"class":517},[507,80752,80753,80756,80758,80760,80762],{"class":509,"line":1379},[507,80754,80755],{"class":517},"counts = job.",[507,80757,23996],{"class":576},[507,80759,13983],{"class":517},[507,80761,73558],{"class":576},[507,80763,781],{"class":517},[507,80765,80766,80769,80771,80773,80776,80778,80781],{"class":509,"line":1389},[507,80767,80768],{"class":517},"best = ",[507,80770,36712],{"class":572},[507,80772,77632],{"class":517},[507,80774,80775],{"class":2155},"key",[507,80777,573],{"class":572},[507,80779,80780],{"class":517},"cut_value)        ",[507,80782,80783],{"class":562},"# best split in the shot pool\n",[507,80785,80786,80788,80790,80792,80795,80797,80800,80803,80805,80808,80810,80812,80814,80816,80818,80820,80823],{"class":509,"line":1397},[507,80787,8525],{"class":572},[507,80789,580],{"class":517},[507,80791,22278],{"class":513},[507,80793,80794],{"class":730},"\"Best cut: ",[507,80796,2810],{"class":583},[507,80798,80799],{"class":576},"cut_value",[507,80801,80802],{"class":517},"(best)",[507,80804,2872],{"class":583},[507,80806,80807],{"class":730}," of ",[507,80809,2810],{"class":583},[507,80811,1763],{"class":572},[507,80813,580],{"class":517},[507,80815,80351],{"class":583},[507,80817,3649],{"class":517},[507,80819,2872],{"class":583},[507,80821,80822],{"class":730}," edges\"",[507,80824,587],{"class":517},[74616,80826,80828],{"lead":80827},"Quantum Courier is open source and MIT-licensed, built to be forked and rerun.",[41852,80829,80830,80838],{},[41855,80831,80832],{},[41858,80833,80834,80836],{},[41861,80835,74627],{},[41861,80837,74630],{},[41868,80839,80840,80847,80855,80863],{},[41858,80841,80842,80844],{},[41873,80843,80076],{},[41873,80845,80846],{},"Vanilla HTML, CSS, and JavaScript, no framework.",[41858,80848,80849,80852],{},[41873,80850,80851],{},"Classical solvers",[41873,80853,80854],{},"Hungarian algorithm, 2-opt + simulated annealing, Goemans-Williamson SDP rounding.",[41858,80856,80857,80860],{},[41873,80858,80859],{},"Quantum-inspired",[41873,80861,80862],{},"QUBO simulated quantum annealing, in the browser.",[41858,80864,80865,80868],{},[41873,80866,80867],{},"Real hardware",[41873,80869,80870],{},"Qiskit + qiskit-ionq, IonQ Forte via qBraid, QAOA p=1 at 4,096 shots.",[13,80872,80874],{"id":80873},"structure-beats-qubit-count","Structure beats qubit count",[18,80876,80877],{},"The most interesting result is the one that went the other way. For Stage 3, Siti tested four QAOA variants on Forte against classical simulated annealing on a 25-customer vehicle-routing instance with time windows. Classical won every time, and she left that result in the game on purpose.",[72443,80879,80880],{"avatar":80193,"name":74256,"role":74353,"username":74354},[18,80881,74357],{},[18,80883,80884],{},"The reason is structure, not raw qubit count. MaxCut has a cost function that simply counts cut edges, which lines up naturally with what a shallow parameterised quantum circuit can express. Vehicle routing does not have that property at the scales we can run today, and classical routing heuristics are decades mature. That contrast is the real lesson Quantum Courier teaches: where quantum helps is a question of problem shape, and the honest answer is sometimes no. Siti is already extending the work toward grid-storage siting problems, with a finding written up for a 2026 submission.",[13,80886,73027],{"id":73026},[18,80888,80889],{},"The game is open and forkable on Qollab, the full code is MIT-licensed on GitHub, and you can play all five stages in your browser right now. The Stage 4 MaxCut circuit runs on real IonQ hardware in a click, so you can rerun the result that beat the classical baseline yourself.",[72443,80891,80892],{"avatar":80193,"name":74256,"role":74353,"username":74354},[18,80893,80894],{},"I hope this doesn't stop after Qollab and IonQ. I'd love to see more collaborations like this between industry and quantum.",[73026,80896,80899],{"fork-href":80286,"live-href":80897,"title":80898},"https:\u002F\u002Fqatalyst-quantum.co.uk\u002Fplay","Can you beat a quantum computer at route planning?",[18,80900,80901,80902],{},"Play all five stages, fork the game, and rerun the MaxCut circuit on real IonQ Forte. ",[154,80903,73040],{},[953,80905,78378],{},{"title":104,"searchDepth":105,"depth":105,"links":80907},[80908,80909,80910,80911],{"id":80199,"depth":105,"text":80200},{"id":80274,"depth":105,"text":80275},{"id":80873,"depth":105,"text":80874},{"id":73026,"depth":105,"text":73027},[112,969,74245],[80914],{"username":74354,"name":74256,"role":80915,"avatar":80193,"bio":80916,"links":80917},"Founder · Qatalyst Quantum","Siti is a postdoctoral researcher at Heriot-Watt University and the founder of Qatalyst Quantum, a vehicle-routing optimisation startup combining classical and quantum approaches (Conception X, Microsoft Founders Hub, Quantinuum Q-NET, Kipu Quantum Hub). She spent two years at the Port of Dover as a KTP Associate, building traffic-simulation models and exploring quantum computing for routing operations, and has built Qatalyst's full optimisation pipeline, from agentic AI orchestration to custom solvers and quantum problem reformulation, with hands-on experience on D-Wave and ORCA hardware.",[80918,80920,80921,80923],{"label":73068,"href":80919},"https:\u002F\u002Fqollab.xyz\u002Fu\u002FSitifar",{"label":73059,"href":79119},{"label":979,"href":80922},"https:\u002F\u002Fgithub.com\u002Fsitifariya",{"label":80924,"href":80925},"Qatalyst ↗","https:\u002F\u002Fqatalyst-quantum.co.uk\u002F",{"username":8,"name":73075,"role":73076,"avatar":73077},"Dr Siti Fariya built a browser game that races classical and quantum solvers across five real logistics problems, and shows honestly which one wins, and why.","Dr Siti Fariya built Quantum Courier, a browser game racing classical vs quantum solvers across five logistics problems, with a 46.2% MaxCut win on IonQ Forte.",{"href":80286,"label":80157},{"image":80262,"alt":80931,"liveUrl":80897},"Quantum Courier: Pizza Race, a browser game racing classical vs quantum solvers",{},"\u002Fblog\u002Fquantum-courier","2026-05-02","7 min read",[],[80938,80939,80940],{"username":74390,"project":77806,"title":74277,"category":73752,"thumb":77807,"to":74383},{"username":72430,"project":73748,"title":73052,"category":1007,"thumb":73749,"to":73750},{"username":73094,"project":73095,"title":73096,"category":73097,"thumb":73098,"to":73099},{"title":80942,"description":80943},"Quantum Creative Project Showcase: Quantum Courier","A browser game that races classical vs quantum solvers across five logistics problems, and shows honestly which wins and why.","blog\u002Fquantum-courier",[75068,143,80174],"vKX5WNXRiWNT2DtNBtSdf0UgvkASzpcM-Yd0SQJ21aA",{"id":80948,"title":73788,"authors":80949,"body":80950,"breadcrumb":81055,"builders":81057,"byline":116,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":81058,"description":81059,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":81060,"heroImage":116,"kind":116,"lessonCount":116,"meta":81063,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":81064,"publishDate":81065,"readingTime":116,"related":81066,"relatedProjects":116,"seo":81067,"stem":81070,"tags":81071,"track":116,"trackName":116,"__hash__":81074},"blog\u002Fblog\u002Fambassadors.md",[8],{"type":10,"value":80951,"toc":81047},[80952,80956,80959,80963,80977,80981,80989,80993,80996,81000,81003,81007,81014,81020,81026,81032,81038],[13,80953,80955],{"id":80954},"bring-quantum-to-your-corner-of-the-world","Bring quantum to your corner of the world",[18,80957,80958],{},"Quantum still feels closed off to most people who could be building with it. Ambassadors change that locally. You already have a community, a campus, a meetup, a Discord, a lab, and you help the people in it take their first real step into quantum. We give you the credits, the tools, and the backup to make that easy.",[13,80960,80962],{"id":80961},"what-ambassadors-do","What ambassadors do",[42,80964,80965,80968,80971,80974],{},[45,80966,80967],{},"Run meetups, workshops, or hack nights where people write and run their first quantum circuit.",[45,80969,80970],{},"Help newcomers get past the intimidating part and onto real hardware through the Qollab Playground.",[45,80972,80973],{},"Share what people in your community are building, so their work gets seen.",[45,80975,80976],{},"Send signal back to the team on what is confusing, missing, or working well.",[13,80978,80980],{"id":80979},"what-you-get","What you get",[29,80982],{"b1":80983,"b2":80984,"b3":80985,"t1":80986,"t2":80987,"t3":80988},"Credits to run on real IonQ quantum hardware, for you and the people you bring in.","New Playground features and templates before they go wide, plus a direct line to the team.","Materials, guidance, and a network of other ambassadors to run great sessions.","Monthly hardware credits","Early access","Event support",[13,80990,80992],{"id":80991},"who-we-are-looking-for","Who we are looking for",[18,80994,80995],{},"You do not need to be a quantum expert. You need to be good at gathering people. Student leaders, community organizers, educators, and developer advocates make the best ambassadors. If you already get people in a room around code, science, or creative work, you can do this.",[13,80997,80999],{"id":80998},"how-to-apply","How to apply",[18,81001,81002],{},"Tell us about your community and what you would like to run. Applications are reviewed on a rolling basis, so there is no deadline to race. If it is a fit, the team follows up to get you set up with credits and materials.",[13,81004,81006],{"id":81005},"frequently-asked-questions","Frequently asked questions",[81008,81009,81011],"faq-item",{"q":81010},"What does a Qollab Ambassador do?",[18,81012,81013],{},"Ambassadors champion quantum computing where they already have a community. They run meetups and workshops, help newcomers get their first circuit running, and share what people in their network are building.",[81008,81015,81017],{"q":81016},"What do ambassadors get?",[18,81018,81019],{},"Monthly credits to run on real IonQ hardware, early access to new Qollab features, a direct line to the team, and support for the events you run.",[81008,81021,81023],{"q":81022},"Who should apply?",[18,81024,81025],{},"Student leaders, community organizers, educators, and developer advocates who already gather people around code, science, or creative work and want to bring quantum into the room.",[81008,81027,81029],{"q":81028},"Do I need to be a quantum expert?",[18,81030,81031],{},"No. You need to be good at bringing people together. We give you the tools, templates, and support to help your community get started with quantum.",[81008,81033,81035],{"q":81034},"How do I apply to the ambassador program?",[18,81036,81037],{},"Applications are rolling. Tell us about your community and what you would like to run, and the team will follow up.",[94,81039,81044],{"dark":104,"f1":81040,"f2":74443,"l1":81041,"l2":81042,"title":81043},"mailto:hello@qollab.xyz?subject=Qollab Ambassador Program","Apply to be an ambassador","Explore Qollab","Ready to lead?",[18,81045,81046],{},"Bring quantum to the people around you. Tell us about your community and we will help you get them building.",{"title":104,"searchDepth":105,"depth":105,"links":81048},[81049,81050,81051,81052,81053,81054],{"id":80954,"depth":105,"text":80955},{"id":80961,"depth":105,"text":80962},{"id":80979,"depth":105,"text":80980},{"id":80991,"depth":105,"text":80992},{"id":80998,"depth":105,"text":80999},{"id":81005,"depth":105,"text":81006},[112,969,81056],"Ambassadors",[],"Champion quantum in your community, help newcomers run their first circuit, and earn monthly IonQ hardware credits while you do it.","Champion quantum computing in your community, help newcomers run their first circuit, and earn monthly IonQ hardware credits. Applications are rolling.",{"primaryHref":81040,"primaryLabel":81061,"secondaryHref":74443,"secondaryLabel":81062},"Apply to the program","See the community →",{},"\u002Fblog\u002Fambassadors","2026-05-01",[],{"title":81068,"description":81069},"Qollab Ambassador Program: bring quantum to your community","Run meetups, help newcomers get their first circuit on real hardware, and earn monthly IonQ credits as a Qollab Ambassador. Rolling applications.","blog\u002Fambassadors",[81072,81073],"program","community","h9CoC7ebGym0MORYWFLB5pCcvkbbDkj_WZmcdbdk670",{"id":81076,"title":81077,"authors":81078,"body":81080,"breadcrumb":81683,"builders":81684,"byline":81704,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":81705,"description":81706,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":81707,"hero":81708,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":81709,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":81710,"publishDate":81711,"readingTime":80935,"related":81712,"relatedProjects":81713,"seo":81717,"stem":81720,"tags":81721,"track":116,"trackName":116,"__hash__":81722},"blog\u002Fblog\u002Fmusiq.md","Project Showcase: Musiq",[73101,81079],"emmanuellaadams",{"type":10,"value":81081,"toc":81676},[81082,81085,81088,81091,81095,81099,81102,81157,81160,81164,81167,81170,81577,81629,81633,81636,81640,81643,81645,81648,81652,81655,81659,81661,81664,81673],[18,81083,81084],{},"Musiq starts from a simple question: what does a quantum circuit sound like?",[18,81086,81087],{},"It is a browser-based quantum sonification studio. You build a circuit out of gates, or import an OpenQASM template, run it on a simulator or real IonQ hardware, and Musiq turns the resulting probabilities, amplitudes, and phases into sound you can play, visualize, and export. Quantum mechanics is usually met through equations and probability distributions; Musiq adds another sense.",[18,81089,81090],{},"It is a Spring 2026 challenge project from Tomoya Hatanaka, a freelance quantum engineer, and Emmanuella Adams, a creative technologist. The starting point was a frustration with how generative music usually works.",[72443,81092,81093],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,81094,75279],{},[13,81096,81098],{"id":81097},"what-a-circuit-sounds-like","What a circuit sounds like",[18,81100,81101],{},"The core of Musiq is a direct mapping from quantum data to audio. After a circuit runs, its outputs are not just plotted, they are turned into the parameters of a sound. Each part of the quantum result drives a part of what you hear:",[74616,81103,81105],{"lead":81104},"Quantum data becomes sound through a direct mapping, computed from each run.",[41852,81106,81107,81115],{},[41855,81108,81109],{},[41858,81110,81111,81113],{},[41861,81112,74627],{},[41861,81114,74630],{},[41868,81116,81117,81125,81133,81141,81149],{},[41858,81118,81119,81122],{},[41873,81120,81121],{},"Basis-state index",[41873,81123,81124],{},"Musical frequency, which note each outcome plays.",[41858,81126,81127,81130],{},[41873,81128,81129],{},"Measurement probability",[41873,81131,81132],{},"Strength of that frequency component.",[41858,81134,81135,81138],{},[41873,81136,81137],{},"Statevector amplitude",[41873,81139,81140],{},"Loudness contribution.",[41858,81142,81143,81146],{},[41873,81144,81145],{},"Statevector phase",[41873,81147,81148],{},"Oscillator phase and interference.",[41858,81150,81151,81154],{},[41873,81152,81153],{},"Quantum distribution",[41873,81155,81156],{},"Overall spectral and tonal texture.",[18,81158,81159],{},"Because the whole distribution shapes the waveform, different circuits sound genuinely different. A Bell or GHZ state's correlated outcomes shift the balance of frequencies. Interference-heavy IQP circuits produce dense, irregular textures with sharp peaks and valleys. Quantum-walk templates spread amplitude across many states and drift through evolving, probability-weighted tones.",[13,81161,81163],{"id":81162},"hearing-a-bell-state","Hearing a Bell state",[18,81165,81166],{},"The simplest thing you can hear is a Bell state: two qubits put into superposition and entangled, so their measurement outcomes are correlated. On Qollab the circuit runs on a simulator or a real QPU exactly as written, and the counts that come back are what Musiq turns into sound.",[831,81168],{"caption":81169,"no":835,"poster":75283,"video":75284},"Building a circuit in the studio, running it, and hearing and seeing the result, with waveform and spectrum. Press play, sound on.",[493,81171,81174],{"name":81172,"run-href":81173,"tag":496},"bell_state.py","\u002Fu\u002Fdoraking\u002Fmusiq",[498,81175,81177],{"className":500,"code":81176,"language":502,"meta":104,"style":104},"# 'backend' is pre-created as a global on Qollab.\nfrom qiskit import QuantumCircuit\n\n# A Bell state: superposition, then entanglement.\ncircuit = QuantumCircuit(2, 2)   # 2 qubits, 2 bits\ncircuit.h(0)\ncircuit.cx(0, 1)\ncircuit.measure([0, 1], [0, 1])\nprint(circuit)\n\nfrom qiskit.providers.jobstatus import JobStatus\nimport time\n\ndef main(shots=100, exclude_low_probability=True, low_threshold=0.05):\n    job = backend.run(circuit, shots=shots)\n\n    # Poll until the job finishes\n    while True:\n        status = job.status()\n        print(f\"Job status is {status}\")\n        if status is JobStatus.DONE:\n            break\n        time.sleep(10)\n\n    counts = job.get_counts()\n    if exclude_low_probability:\n        threshold = shots * low_threshold\n        # Filter low-probability noise\n        counts = {b: c for b, c in counts.items() if c > threshold}\n    print(f\"Counts for {shots} shots: {counts}\")\n",[504,81178,81179,81184,81194,81198,81203,81224,81243,81266,81290,81297,81301,81311,81317,81321,81355,81378,81382,81387,81395,81408,81429,81444,81449,81461,81465,81482,81489,81503,81508,81546],{"__ignoreMap":104},[507,81180,81181],{"class":509,"line":510},[507,81182,81183],{"class":562},"# 'backend' is pre-created as a global on Qollab.\n",[507,81185,81186,81188,81190,81192],{"class":509,"line":105},[507,81187,529],{"class":513},[507,81189,532],{"class":517},[507,81191,514],{"class":513},[507,81193,537],{"class":517},[507,81195,81196],{"class":509,"line":540},[507,81197,556],{"emptyLinePlaceholder":133},[507,81199,81200],{"class":509,"line":553},[507,81201,81202],{"class":562},"# A Bell state: superposition, then entanglement.\n",[507,81204,81205,81207,81209,81211,81213,81215,81217,81219,81221],{"class":509,"line":559},[507,81206,73339],{"class":517},[507,81208,573],{"class":572},[507,81210,577],{"class":576},[507,81212,580],{"class":517},[507,81214,584],{"class":583},[507,81216,622],{"class":517},[507,81218,584],{"class":583},[507,81220,67189],{"class":517},[507,81222,81223],{"class":562},"# 2 qubits, 2 bits\n",[507,81225,81226,81228,81230,81232,81234,81236],{"class":509,"line":566},[507,81227,73358],{"class":517},[507,81229,596],{"class":576},[507,81231,580],{"class":517},[507,81233,601],{"class":583},[507,81235,3649],{"class":517},[507,81237,72708,81238],{"class":72706,"tabindex":72707},[507,81239,81240,81242],{"class":72711,"role":72712},[154,81241,73373],{}," A Hadamard spreads the qubit across 0 and 1. In Musiq, that spread becomes the range of frequencies you hear.",[507,81244,81245,81247,81249,81251,81253,81255,81257,81259],{"class":509,"line":590},[507,81246,73358],{"class":517},[507,81248,615],{"class":576},[507,81250,580],{"class":517},[507,81252,601],{"class":583},[507,81254,622],{"class":517},[507,81256,625],{"class":583},[507,81258,3649],{"class":517},[507,81260,72708,81261],{"class":72706,"tabindex":72707},[507,81262,81263,81265],{"class":72711,"role":72712},[154,81264,73431],{}," A CNOT links the two qubits, so their outcomes are correlated. That reshapes the distribution, and so the balance of frequencies in the sound.",[507,81267,81268,81270,81272,81274,81276,81278,81280,81282,81284,81286,81288],{"class":509,"line":610},[507,81269,73358],{"class":517},[507,81271,72822],{"class":576},[507,81273,79780],{"class":517},[507,81275,601],{"class":583},[507,81277,622],{"class":517},[507,81279,625],{"class":583},[507,81281,75921],{"class":517},[507,81283,601],{"class":583},[507,81285,622],{"class":517},[507,81287,625],{"class":583},[507,81289,68725],{"class":517},[507,81291,81292,81294],{"class":509,"line":634},[507,81293,8525],{"class":572},[507,81295,81296],{"class":517},"(circuit)\n",[507,81298,81299],{"class":509,"line":661},[507,81300,556],{"emptyLinePlaceholder":133},[507,81302,81303,81305,81307,81309],{"class":509,"line":678},[507,81304,529],{"class":513},[507,81306,73222],{"class":517},[507,81308,514],{"class":513},[507,81310,73227],{"class":517},[507,81312,81313,81315],{"class":509,"line":683},[507,81314,514],{"class":513},[507,81316,80323],{"class":517},[507,81318,81319],{"class":509,"line":697},[507,81320,556],{"emptyLinePlaceholder":133},[507,81322,81323,81325,81327,81329,81331,81333,81335,81337,81340,81342,81344,81346,81349,81351,81353],{"class":509,"line":710},[507,81324,1370],{"class":513},[507,81326,73467],{"class":576},[507,81328,580],{"class":517},[507,81330,68762],{"class":1382},[507,81332,573],{"class":517},[507,81334,5682],{"class":583},[507,81336,622],{"class":517},[507,81338,81339],{"class":1382},"exclude_low_probability",[507,81341,573],{"class":517},[507,81343,13878],{"class":583},[507,81345,622],{"class":517},[507,81347,81348],{"class":1382},"low_threshold",[507,81350,573],{"class":517},[507,81352,72745],{"class":583},[507,81354,1883],{"class":517},[507,81356,81357,81359,81361,81363,81365,81367,81369,81371,81373],{"class":509,"line":715},[507,81358,73482],{"class":517},[507,81360,573],{"class":572},[507,81362,73487],{"class":517},[507,81364,22501],{"class":576},[507,81366,73492],{"class":517},[507,81368,68762],{"class":2155},[507,81370,573],{"class":572},[507,81372,73499],{"class":517},[507,81374,72708,81375],{"class":72706,"tabindex":72707},[507,81376,81377],{"class":72711,"role":72712},"Submits to a simulator or a real IonQ QPU through Qollab. Audio from a QPU run is derived from measurements on physical hardware.",[507,81379,81380],{"class":509,"line":721},[507,81381,556],{"emptyLinePlaceholder":133},[507,81383,81384],{"class":509,"line":736},[507,81385,81386],{"class":562},"    # Poll until the job finishes\n",[507,81388,81389,81391,81393],{"class":509,"line":748},[507,81390,73513],{"class":513},[507,81392,64764],{"class":583},[507,81394,1728],{"class":517},[507,81396,81397,81400,81402,81404,81406],{"class":509,"line":761},[507,81398,81399],{"class":517},"        status ",[507,81401,573],{"class":572},[507,81403,23993],{"class":517},[507,81405,73518],{"class":576},[507,81407,781],{"class":517},[507,81409,81410,81412,81414,81416,81419,81421,81423,81425,81427],{"class":509,"line":775},[507,81411,64185],{"class":572},[507,81413,580],{"class":517},[507,81415,22278],{"class":513},[507,81417,81418],{"class":730},"\"Job status is ",[507,81420,2810],{"class":583},[507,81422,73518],{"class":517},[507,81424,2872],{"class":583},[507,81426,22281],{"class":730},[507,81428,587],{"class":517},[507,81430,81431,81433,81436,81438,81440,81442],{"class":509,"line":784},[507,81432,1734],{"class":513},[507,81434,81435],{"class":517}," status ",[507,81437,37008],{"class":513},[507,81439,73527],{"class":517},[507,81441,73530],{"class":583},[507,81443,1728],{"class":517},[507,81445,81446],{"class":509,"line":796},[507,81447,81448],{"class":513},"            break\n",[507,81450,81451,81453,81455,81457,81459],{"class":509,"line":809},[507,81452,73537],{"class":517},[507,81454,73540],{"class":576},[507,81456,580],{"class":517},[507,81458,23805],{"class":583},[507,81460,587],{"class":517},[507,81462,81463],{"class":509,"line":1352},[507,81464,556],{"emptyLinePlaceholder":133},[507,81466,81467,81469,81471,81473,81475,81477],{"class":509,"line":1357},[507,81468,73551],{"class":517},[507,81470,573],{"class":572},[507,81472,23993],{"class":517},[507,81474,73558],{"class":576},[507,81476,66172],{"class":517},[507,81478,72708,81479],{"class":72706,"tabindex":72707},[507,81480,81481],{"class":72711,"role":72712},"Each basis state's index becomes a musical frequency; how often it appears sets that frequency's strength.",[507,81483,81484,81486],{"class":509,"line":1362},[507,81485,1717],{"class":513},[507,81487,81488],{"class":517}," exclude_low_probability:\n",[507,81490,81491,81494,81496,81498,81500],{"class":509,"line":1367},[507,81492,81493],{"class":517},"        threshold ",[507,81495,573],{"class":572},[507,81497,69203],{"class":517},[507,81499,2391],{"class":572},[507,81501,81502],{"class":517}," low_threshold\n",[507,81504,81505],{"class":509,"line":1379},[507,81506,81507],{"class":562},"        # Filter low-probability noise\n",[507,81509,81510,81513,81515,81518,81520,81523,81525,81527,81529,81531,81533,81536,81538,81541],{"class":509,"line":1389},[507,81511,81512],{"class":517},"        counts ",[507,81514,573],{"class":572},[507,81516,81517],{"class":517}," {b: c ",[507,81519,1630],{"class":513},[507,81521,81522],{"class":517}," b, c ",[507,81524,1636],{"class":513},[507,81526,73595],{"class":517},[507,81528,22607],{"class":576},[507,81530,1677],{"class":517},[507,81532,1645],{"class":513},[507,81534,81535],{"class":517}," c ",[507,81537,1651],{"class":572},[507,81539,81540],{"class":517}," threshold}",[507,81542,72708,81543],{"class":72706,"tabindex":72707},[507,81544,81545],{"class":72711,"role":72712},"Drops outcomes too rare to be signal, a simple hardware-noise filter, before the counts are mapped to sound.",[507,81547,81548,81550,81552,81554,81557,81559,81561,81563,81566,81568,81571,81573,81575],{"class":509,"line":1397},[507,81549,2060],{"class":572},[507,81551,580],{"class":517},[507,81553,22278],{"class":513},[507,81555,81556],{"class":730},"\"Counts for ",[507,81558,2810],{"class":583},[507,81560,68762],{"class":517},[507,81562,2872],{"class":583},[507,81564,81565],{"class":730}," shots: ",[507,81567,2810],{"class":583},[507,81569,81570],{"class":517},"counts",[507,81572,2872],{"class":583},[507,81574,22281],{"class":730},[507,81576,587],{"class":517},[74616,81578,81580],{"lead":81579},"Musiq is open source, built to be forked and rerun.",[41852,81581,81582,81590],{},[41855,81583,81584],{},[41858,81585,81586,81588],{},[41861,81587,74627],{},[41861,81589,74630],{},[41868,81591,81592,81599,81606,81613,81621],{},[41858,81593,81594,81596],{},[41873,81595,80076],{},[41873,81597,81598],{},"Web app, visual circuit editor, players, and spectrum views.",[41858,81600,81601,81603],{},[41873,81602,72215],{},[41873,81604,81605],{},"Qiskit and OpenQASM 2.0, Bell, GHZ, IQP, and quantum-walk templates.",[41858,81607,81608,81610],{},[41873,81609,72293],{},[41873,81611,81612],{},"Local ideal simulator, IonQ simulator, or IonQ QPU.",[41858,81614,81615,81618],{},[41873,81616,81617],{},"Audio",[41873,81619,81620],{},"Python with NumPy, SciPy, and SoundFile; WAV output plus spectral analysis.",[41858,81622,81623,81626],{},[41873,81624,81625],{},"Deployment",[41873,81627,81628],{},"Cloud-hosted web app.",[13,81630,81632],{"id":81631},"a-universal-translator","A universal translator",[18,81634,81635],{},"Musiq is built to be a teaching instrument as much as a creative one. By comparing how different circuits sound, a learner can build intuition for how circuit design shapes quantum output, without first having to read the math. That is the role Emmanuella works on: the translation from quantum behavior into an experience a newcomer can actually feel their way through.",[72443,81637,81638],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,81639,75185],{},[18,81641,81642],{},"The team is careful about what the tool is and is not. Musiq is a sonification instrument that complements the usual diagrams and equations rather than replacing them, and it is honest about its current scope: it processes a circuit's output as a whole, and does not yet assign individual qubits to separate voices or instruments. What it does do is give superposition, interference, and entanglement a sound, and a spectrum you can inspect afterward.",[13,81644,73675],{"id":73674},[18,81646,81647],{},"Today Musiq generates a single layered waveform from a circuit, with a set of circuit templates and analysis tools to pick apart the result. The ambition Tomoya describes is bigger: to move from sonification toward genuine composition, using the all-to-all connectivity of trapped-ion hardware to give separate musical voices their own entangled structure.",[72443,81649,81650],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,81651,75254],{},[18,81653,81654],{},"It is also a small argument about what quantum computers are for. Most of the field points its hardware at optimization and simulation; Musiq points it at a speaker.",[72443,81656,81657],{"avatar":104,"name":75181,"role":75182,"username":73101},[18,81658,75292],{},[13,81660,73027],{"id":73026},[18,81662,81663],{},"Musiq is open and forkable on Qollab, the full studio is on GitHub, and you can build a circuit and hear it in your browser right now. Start from a Bell or GHZ template, or bring your own OpenQASM, and listen to how the design changes the sound.",[73026,81665,81668],{"fork-href":81173,"live-href":81666,"title":81667},"https:\u002F\u002Fmusiquantum.vercel.app\u002F","Build a circuit. Hear what it does.",[18,81669,81670,81671],{},"Fork Musiq, run a circuit on a simulator or real IonQ hardware, and turn its quantum data into sound. ",[154,81672,73040],{},[953,81674,81675],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":81677},[81678,81679,81680,81681,81682],{"id":81097,"depth":105,"text":81098},{"id":81162,"depth":105,"text":81163},{"id":81631,"depth":105,"text":81632},{"id":73674,"depth":105,"text":73675},{"id":73026,"depth":105,"text":73027},[112,969,73103],[81685,81697],{"username":73101,"name":75181,"role":81686,"avatar":104,"bio":81687,"links":81688},"Project lead · quantum researcher & engineer","Tomoya is a freelance quantum engineer with an Applied Physics master's from the University of Tokyo, specializing in quantum error correction and quantum algorithms, including ML-based decoders and quantum hash functions. He is first author on a published quantum hash function paper with RIKEN and OIST co-authors, and founded the open-source project KetQat to help democratize quantum computing.",[81689,81691,81692,81694],{"label":73068,"href":81690},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fdoraking",{"label":73059,"href":79286},{"label":979,"href":81693},"https:\u002F\u002Fgithub.com\u002Fdorakingx",{"label":81695,"href":81696},"arXiv ↗","https:\u002F\u002Farxiv.org\u002Fabs\u002F2409.19932",{"username":81079,"name":79291,"role":81698,"avatar":104,"bio":81699,"links":81700},"Creative technologist · quantum experience","Emmanuella works on the bridge between quantum systems and creative research, focused on translating complex technical concepts into human-centered experiences. On Musiq she shapes the quantum-to-music translation, the audiovisual experience, and the educational interaction layer. She is an undergraduate in software engineering at Federal University Dutse, specializing in AI and machine learning.",[81701,81702],{"label":73059,"href":79290},{"label":979,"href":81703},"https:\u002F\u002Fgithub.com\u002FEmmanuella-Adams",{"username":8,"name":73075,"role":73076,"avatar":73077},"Tomoya Hatanaka and Emmanuella Adams built a browser studio that turns quantum circuits into sound. Build a circuit, run it on a simulator or real IonQ hardware, and hear its quantum data become music.","Tomoya Hatanaka and Emmanuella Adams built Musiq, a browser studio that turns quantum circuits into sound on real IonQ hardware. A Qollab Spring 2026 project.",{"href":81173,"label":75367},{"image":75283,"alt":75282,"liveUrl":81666},{},"\u002Fblog\u002Fmusiq","2026-04-30",[],[81714,81715,81716],{"username":73094,"project":73095,"title":73096,"category":73097,"thumb":73098,"to":73099},{"username":74390,"project":77806,"title":74277,"category":73752,"thumb":77807,"to":74383},{"username":74372,"project":79092,"title":74261,"category":73752,"thumb":79093,"to":74365},{"title":81718,"description":81719},"Quantum Creative Project Showcase: Musiq","A browser studio that turns quantum circuits into sound. Build a circuit, run it on IonQ, and hear the quantum data become music.","blog\u002Fmusiq",[73758,143,74477],"Crb77p51Xb7WAYdrxu4wfDnUnVkr2DdtpE420l0SJBA",{"id":81724,"title":81725,"authors":81726,"body":81727,"breadcrumb":82583,"builders":82584,"byline":82596,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":82597,"description":82598,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":82599,"hero":82601,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":82603,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":82604,"publishDate":82605,"readingTime":74132,"related":82606,"relatedProjects":82607,"seo":82611,"stem":82614,"tags":82615,"track":116,"trackName":116,"__hash__":82616},"blog\u002Fblog\u002Fquantum-butterfly-field.md","Project Showcase: Quantum Butterfly Field",[1011],{"type":10,"value":81728,"toc":82576},[81729,81732,81739,81746,81750,81754,81762,81765,81769,81772,81776,81779,81782,81785,82455,82458,82504,82508,82511,82531,82542,82546,82553,82557,82560,82562,82565,82574],[18,81730,81731],{},"Quantum Butterfly Field is an interactive artwork where five butterflies are five qubits. As the circuit runs, their individual identities dissolve into a single entangled field.",[18,81733,81734,81735,81738],{},"Then one butterfly is damaged, severed from the whole. In a classical world that loss would be final. Here it is not: through the quantum ",[154,81736,81737],{},"anti-butterfly effect",", what was lost is recovered from the deeply entangled correlations that still bind the field together. The information was never in that one butterfly alone.",[18,81740,81741,81742,81745],{},"It is a Spring 2026 challenge project from Xinyi Zhang, a multidisciplinary artist and technologist who builds at the seam between physics and feeling. She frames the piece through the Native Hawaiian concept of ",[1031,81743,81744],{},"lōkahi",", where the wellbeing of any one part depends on the integrity of the whole web it belongs to.",[72443,81747,81748],{"avatar":73934,"name":73880,"role":73935,"username":1011},[18,81749,73938],{},[13,81751,81753],{"id":81752},"the-anti-butterfly-effect","The anti-butterfly effect",[18,81755,81756,81757,81761],{},"The piece is built on a counterintuitive result from quantum information theory, the paper ",[49,81758,81760],{"href":81759},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2003.07267","Recovery of Damaged Information and the Out-of-Time-Ordered Correlators"," (Yan & Sinitsyn, 2020). In a classical chaotic system, small damage cascades into large changes: a butterfly flaps its wings and a tornado follows. In a quantum system, this is not the case.",[18,81763,81764],{},"Once information has been scrambled deeply enough across an entangled system, a local disturbance cannot destroy it. The information no longer lives in any single qubit, but in the correlations between all of them. By winding the scrambling circuit backward, the damaged qubit's original state is recovered almost completely, marked only by a small residual trace. The authors called this the anti-butterfly effect: at the quantum scale, reality is self-healing.",[72443,81766,81767],{"avatar":73934,"name":73880,"role":74038,"username":1011},[18,81768,74041],{},[18,81770,81771],{},"That question is the whole brief. The artwork does not explain the physics so much as stage it, turning an abstract theorem about scrambling and recovery into something you watch happen to a field of living things.",[13,81773,81775],{"id":81774},"five-butterflies-one-field","Five butterflies, one field",[18,81777,81778],{},"Underneath the animation is a real five-qubit scrambling circuit, and the metaphor maps onto it exactly. Each butterfly is a qubit. When the butterflies dance together, the gates entangle them, and each one's state is spread across the whole field like a memory held in relationship rather than in any single place.",[18,81780,81781],{},"The protocol runs in four phases, and on Qollab the circuit runs on hardware exactly as written:",[831,81783],{"caption":81784,"no":835,"poster":74034,"video":74035},"The five-butterfly field dancing through a scrambling circuit, entangling as the protocol runs. Press play, sound on.",[493,81786,81789],{"name":81787,"run-href":81788,"tag":72511},"qbf_protocol.py","\u002Fu\u002Fxinyi\u002Fquantum-butterfly-field",[498,81790,81792],{"className":500,"code":81791,"language":502,"meta":72515,"style":104},"# 'backend' is pre-created from the \"Select QPU\" dropdown below.\nfrom qiskit import QuantumCircuit, transpile\nfrom qiskit.providers.jobstatus import JobStatus\nimport numpy as np, time\n\nN_QUBITS, N_LAYERS, SHOTS = 5, 3, 1000   # five butterflies, three scrambling layers\n\ndef scramble(n, n_layers, seed):\n    # A reproducible random unitary U — a \"fast scrambler\".\n    rng = np.random.default_rng(seed); layers = []\n    for _ in range(n_layers):\n        layer  = [('rx', float(rng.uniform(0, 2*np.pi)), q) for q in range(n)]\n        layer += [('rz', float(rng.uniform(0, 2*np.pi)), q) for q in range(n)]\n        qubits = list(range(n)); rng.shuffle(qubits)        # random all-to-all pairs\n        layer += [('cx', qubits[i], qubits[i+1]) for i in range(0, n-1, 2)]\n        layers.append(layer)\n    return layers\n\ndef build(damaged, seed):\n    layers  = scramble(N_QUBITS, N_LAYERS, seed)\n    ancilla = N_QUBITS\n    qc = QuantumCircuit(N_QUBITS + 1, N_QUBITS)\n\n    for q in range(N_QUBITS):                  # 1. INIT\n        if q != damaged: qc.h(q)\n\n    apply_gates(qc, layers)                    # 2. SCRAMBLE: identities dissolve into one field\n\n    qc.h(ancilla); qc.cx(ancilla, damaged)    # 3. DAMAGE\n\n    apply_inverse_gates(qc, layers)            # 4. HEAL: run the scramble backward (U†)\n\n    qc.measure(range(N_QUBITS), range(N_QUBITS))\n    return qc\n\nqc  = transpile(build(damaged=2, seed=42), backend, optimization_level=1)\njob = backend.run(qc, shots=SHOTS)\nwhile job.status() is not JobStatus.DONE:\n    time.sleep(5)\ncounts = job.result().get_counts()   # tomography in Z, X, Y -> fidelity of the healed butterfly\n",[504,81793,81794,81798,81808,81818,81828,81832,81859,81863,81886,81891,81909,81922,81968,82010,82034,82080,82090,82097,82101,82119,82139,82149,82171,82175,82195,82219,82223,82242,82246,82271,82275,82298,82302,82326,82332,82336,82378,82403,82423,82435],{"__ignoreMap":104},[507,81795,81796],{"class":509,"line":510},[507,81797,79555],{"class":562},[507,81799,81800,81802,81804,81806],{"class":509,"line":105},[507,81801,529],{"class":513},[507,81803,532],{"class":517},[507,81805,514],{"class":513},[507,81807,77077],{"class":517},[507,81809,81810,81812,81814,81816],{"class":509,"line":540},[507,81811,529],{"class":513},[507,81813,73222],{"class":517},[507,81815,514],{"class":513},[507,81817,73227],{"class":517},[507,81819,81820,81822,81824,81826],{"class":509,"line":553},[507,81821,514],{"class":513},[507,81823,518],{"class":517},[507,81825,521],{"class":513},[507,81827,79586],{"class":517},[507,81829,81830],{"class":509,"line":559},[507,81831,556],{"emptyLinePlaceholder":133},[507,81833,81834,81836,81838,81840,81842,81844,81846,81848,81850,81852,81854,81856],{"class":509,"line":566},[507,81835,76387],{"class":583},[507,81837,622],{"class":517},[507,81839,76397],{"class":583},[507,81841,622],{"class":517},[507,81843,73263],{"class":583},[507,81845,1423],{"class":572},[507,81847,9001],{"class":583},[507,81849,622],{"class":517},[507,81851,8226],{"class":583},[507,81853,622],{"class":517},[507,81855,79870],{"class":583},[507,81857,81858],{"class":562},"   # five butterflies, three scrambling layers\n",[507,81860,81861],{"class":509,"line":590},[507,81862,556],{"emptyLinePlaceholder":133},[507,81864,81865,81867,81870,81872,81874,81876,81879,81881,81884],{"class":509,"line":610},[507,81866,1370],{"class":513},[507,81868,81869],{"class":576}," scramble",[507,81871,580],{"class":517},[507,81873,4420],{"class":1382},[507,81875,622],{"class":517},[507,81877,81878],{"class":1382},"n_layers",[507,81880,622],{"class":517},[507,81882,81883],{"class":1382},"seed",[507,81885,1883],{"class":517},[507,81887,81888],{"class":509,"line":634},[507,81889,81890],{"class":562},"    # A reproducible random unitary U — a \"fast scrambler\".\n",[507,81892,81893,81896,81898,81900,81902,81905,81907],{"class":509,"line":661},[507,81894,81895],{"class":517},"    rng ",[507,81897,573],{"class":572},[507,81899,68710],{"class":517},[507,81901,68713],{"class":576},[507,81903,81904],{"class":517},"(seed); layers ",[507,81906,573],{"class":572},[507,81908,1910],{"class":517},[507,81910,81911,81913,81915,81917,81919],{"class":509,"line":678},[507,81912,1916],{"class":513},[507,81914,8216],{"class":517},[507,81916,1636],{"class":513},[507,81918,8221],{"class":572},[507,81920,81921],{"class":517},"(n_layers):\n",[507,81923,81924,81927,81929,81931,81934,81936,81938,81941,81944,81946,81948,81950,81952,81954,81957,81959,81961,81963,81965],{"class":509,"line":683},[507,81925,81926],{"class":517},"        layer  ",[507,81928,573],{"class":572},[507,81930,80356],{"class":517},[507,81932,81933],{"class":730},"'rx'",[507,81935,622],{"class":517},[507,81937,1406],{"class":572},[507,81939,81940],{"class":517},"(rng.",[507,81942,81943],{"class":576},"uniform",[507,81945,580],{"class":517},[507,81947,601],{"class":583},[507,81949,622],{"class":517},[507,81951,584],{"class":583},[507,81953,2391],{"class":572},[507,81955,81956],{"class":517},"np.pi)), q) ",[507,81958,1630],{"class":513},[507,81960,22114],{"class":517},[507,81962,1636],{"class":513},[507,81964,8221],{"class":572},[507,81966,81967],{"class":517},"(n)]\n",[507,81969,81970,81973,81975,81977,81980,81982,81984,81986,81988,81990,81992,81994,81996,81998,82000,82002,82004,82006,82008],{"class":509,"line":697},[507,81971,81972],{"class":517},"        layer ",[507,81974,2285],{"class":572},[507,81976,80356],{"class":517},[507,81978,81979],{"class":730},"'rz'",[507,81981,622],{"class":517},[507,81983,1406],{"class":572},[507,81985,81940],{"class":517},[507,81987,81943],{"class":576},[507,81989,580],{"class":517},[507,81991,601],{"class":583},[507,81993,622],{"class":517},[507,81995,584],{"class":583},[507,81997,2391],{"class":572},[507,81999,81956],{"class":517},[507,82001,1630],{"class":513},[507,82003,22114],{"class":517},[507,82005,1636],{"class":513},[507,82007,8221],{"class":572},[507,82009,81967],{"class":517},[507,82011,82012,82015,82017,82019,82021,82023,82026,82028,82031],{"class":509,"line":710},[507,82013,82014],{"class":517},"        qubits ",[507,82016,573],{"class":572},[507,82018,22078],{"class":572},[507,82020,580],{"class":517},[507,82022,2204],{"class":572},[507,82024,82025],{"class":517},"(n)); rng.",[507,82027,73637],{"class":576},[507,82029,82030],{"class":517},"(qubits)        ",[507,82032,82033],{"class":562},"# random all-to-all pairs\n",[507,82035,82036,82038,82040,82042,82045,82048,82050,82052,82054,82056,82058,82060,82062,82064,82066,82069,82071,82073,82075,82077],{"class":509,"line":715},[507,82037,81972],{"class":517},[507,82039,2285],{"class":572},[507,82041,80356],{"class":517},[507,82043,82044],{"class":730},"'cx'",[507,82046,82047],{"class":517},", qubits[i], qubits[i",[507,82049,2107],{"class":572},[507,82051,625],{"class":583},[507,82053,65188],{"class":517},[507,82055,1630],{"class":513},[507,82057,8246],{"class":517},[507,82059,1636],{"class":513},[507,82061,8221],{"class":572},[507,82063,580],{"class":517},[507,82065,601],{"class":583},[507,82067,82068],{"class":517},", n",[507,82070,2367],{"class":572},[507,82072,625],{"class":583},[507,82074,622],{"class":517},[507,82076,584],{"class":583},[507,82078,82079],{"class":517},")]\n",[507,82081,82082,82085,82087],{"class":509,"line":721},[507,82083,82084],{"class":517},"        layers.",[507,82086,1939],{"class":576},[507,82088,82089],{"class":517},"(layer)\n",[507,82091,82092,82094],{"class":509,"line":736},[507,82093,2504],{"class":513},[507,82095,82096],{"class":517}," layers\n",[507,82098,82099],{"class":509,"line":748},[507,82100,556],{"emptyLinePlaceholder":133},[507,82102,82103,82105,82108,82110,82113,82115,82117],{"class":509,"line":761},[507,82104,1370],{"class":513},[507,82106,82107],{"class":576}," build",[507,82109,580],{"class":517},[507,82111,82112],{"class":1382},"damaged",[507,82114,622],{"class":517},[507,82116,81883],{"class":1382},[507,82118,1883],{"class":517},[507,82120,82121,82124,82126,82128,82130,82132,82134,82136],{"class":509,"line":775},[507,82122,82123],{"class":517},"    layers  ",[507,82125,573],{"class":572},[507,82127,81869],{"class":576},[507,82129,580],{"class":517},[507,82131,76387],{"class":583},[507,82133,622],{"class":517},[507,82135,76397],{"class":583},[507,82137,82138],{"class":517},", seed)\n",[507,82140,82141,82144,82146],{"class":509,"line":784},[507,82142,82143],{"class":517},"    ancilla ",[507,82145,573],{"class":572},[507,82147,82148],{"class":583}," N_QUBITS\n",[507,82150,82151,82153,82155,82157,82159,82161,82163,82165,82167,82169],{"class":509,"line":796},[507,82152,72833],{"class":517},[507,82154,573],{"class":572},[507,82156,577],{"class":576},[507,82158,580],{"class":517},[507,82160,76387],{"class":583},[507,82162,8313],{"class":572},[507,82164,1426],{"class":583},[507,82166,622],{"class":517},[507,82168,76387],{"class":583},[507,82170,587],{"class":517},[507,82172,82173],{"class":509,"line":809},[507,82174,556],{"emptyLinePlaceholder":133},[507,82176,82177,82179,82181,82183,82185,82187,82189,82192],{"class":509,"line":1352},[507,82178,1916],{"class":513},[507,82180,22114],{"class":517},[507,82182,1636],{"class":513},[507,82184,8221],{"class":572},[507,82186,580],{"class":517},[507,82188,76387],{"class":583},[507,82190,82191],{"class":517},"):                  ",[507,82193,82194],{"class":562},"# 1. INIT\n",[507,82196,82197,82199,82201,82203,82206,82208,82211],{"class":509,"line":1357},[507,82198,1734],{"class":513},[507,82200,22114],{"class":517},[507,82202,22619],{"class":572},[507,82204,82205],{"class":517}," damaged: qc.",[507,82207,596],{"class":576},[507,82209,82210],{"class":517},"(q)",[507,82212,72708,82213],{"class":72706,"tabindex":72707},[507,82214,82215,82218],{"class":72711,"role":72712},[154,82216,82217],{},"Init."," Every butterfly starts in superposition except the one to be damaged, which begins in a definite state so its recovery can be measured.",[507,82220,82221],{"class":509,"line":1362},[507,82222,556],{"emptyLinePlaceholder":133},[507,82224,82225,82228,82231,82234],{"class":509,"line":1367},[507,82226,82227],{"class":576},"    apply_gates",[507,82229,82230],{"class":517},"(qc, layers)                    ",[507,82232,82233],{"class":562},"# 2. SCRAMBLE: identities dissolve into one field",[507,82235,72708,82236],{"class":72706,"tabindex":72707},[507,82237,82238,82241],{"class":72711,"role":72712},[154,82239,82240],{},"Scramble."," The unitary U mixes random Rx + Rz rotations with random all-to-all CX pairs: a fast scrambler that spreads each butterfly across the whole field in O(log n) layers.",[507,82243,82244],{"class":509,"line":1379},[507,82245,556],{"emptyLinePlaceholder":133},[507,82247,82248,82250,82252,82255,82257,82260,82263],{"class":509,"line":1389},[507,82249,21867],{"class":517},[507,82251,596],{"class":576},[507,82253,82254],{"class":517},"(ancilla); qc.",[507,82256,615],{"class":576},[507,82258,82259],{"class":517},"(ancilla, damaged)    ",[507,82261,82262],{"class":562},"# 3. DAMAGE",[507,82264,72708,82265],{"class":72706,"tabindex":72707},[507,82266,82267,82270],{"class":72711,"role":72712},[154,82268,82269],{},"Damage."," An ancilla in |+⟩ entangles with the damaged butterfly and is then discarded, severing its correlations with the field. That is the rupture.",[507,82272,82273],{"class":509,"line":1397},[507,82274,556],{"emptyLinePlaceholder":133},[507,82276,82277,82280,82283,82286],{"class":509,"line":1412},[507,82278,82279],{"class":576},"    apply_inverse_gates",[507,82281,82282],{"class":517},"(qc, layers)            ",[507,82284,82285],{"class":562},"# 4. HEAL: run the scramble backward (U†)",[507,82287,72708,82288],{"class":72706,"tabindex":72707},[507,82289,82290,82293,82294,82297],{"class":72711,"role":72712},[154,82291,82292],{},"Heal."," Healing runs the exact scramble backward: reversed order, negated angles. Because the information now lives in the correlations, ",[504,82295,82296],{},"U†"," gathers it back.",[507,82299,82300],{"class":509,"line":1431},[507,82301,556],{"emptyLinePlaceholder":133},[507,82303,82304,82306,82308,82310,82312,82314,82316,82318,82320,82322,82324],{"class":509,"line":1449},[507,82305,21867],{"class":517},[507,82307,72822],{"class":576},[507,82309,580],{"class":517},[507,82311,2204],{"class":572},[507,82313,580],{"class":517},[507,82315,76387],{"class":583},[507,82317,2213],{"class":517},[507,82319,2204],{"class":572},[507,82321,580],{"class":517},[507,82323,76387],{"class":583},[507,82325,22540],{"class":517},[507,82327,82328,82330],{"class":509,"line":1465},[507,82329,2504],{"class":513},[507,82331,72990],{"class":517},[507,82333,82334],{"class":509,"line":1471},[507,82335,556],{"emptyLinePlaceholder":133},[507,82337,82338,82341,82343,82345,82347,82350,82352,82354,82356,82358,82360,82362,82364,82367,82370,82372,82374,82376],{"class":509,"line":1477},[507,82339,82340],{"class":517},"qc  ",[507,82342,573],{"class":572},[507,82344,76717],{"class":576},[507,82346,580],{"class":517},[507,82348,82349],{"class":576},"build",[507,82351,580],{"class":517},[507,82353,82112],{"class":2155},[507,82355,573],{"class":572},[507,82357,584],{"class":583},[507,82359,622],{"class":517},[507,82361,81883],{"class":2155},[507,82363,573],{"class":572},[507,82365,82366],{"class":583},"42",[507,82368,82369],{"class":517},"), backend, ",[507,82371,76734],{"class":2155},[507,82373,573],{"class":572},[507,82375,625],{"class":583},[507,82377,587],{"class":517},[507,82379,82380,82382,82384,82386,82388,82390,82392,82394,82396,82398],{"class":509,"line":1482},[507,82381,23964],{"class":517},[507,82383,573],{"class":572},[507,82385,73487],{"class":517},[507,82387,22501],{"class":576},[507,82389,79863],{"class":517},[507,82391,68762],{"class":2155},[507,82393,573],{"class":572},[507,82395,73263],{"class":583},[507,82397,3649],{"class":517},[507,82399,72708,82400],{"class":72706,"tabindex":72707},[507,82401,82402],{"class":72711,"role":72712},"Submits to IonQ Forte through Qollab. The recovered state's fidelity (0.5 = lost, 1.0 = fully healed) drives the damaged butterfly's luminosity in the artwork.",[507,82404,82405,82407,82409,82411,82413,82415,82417,82419,82421],{"class":509,"line":1488},[507,82406,79882],{"class":513},[507,82408,23993],{"class":517},[507,82410,73518],{"class":576},[507,82412,1677],{"class":517},[507,82414,37008],{"class":513},[507,82416,21980],{"class":513},[507,82418,73527],{"class":517},[507,82420,73530],{"class":583},[507,82422,1728],{"class":517},[507,82424,82425,82427,82429,82431,82433],{"class":509,"line":1494},[507,82426,79903],{"class":517},[507,82428,73540],{"class":576},[507,82430,580],{"class":517},[507,82432,58245],{"class":583},[507,82434,587],{"class":517},[507,82436,82437,82439,82441,82443,82445,82447,82449,82452],{"class":509,"line":1500},[507,82438,79916],{"class":517},[507,82440,573],{"class":572},[507,82442,23993],{"class":517},[507,82444,23996],{"class":576},[507,82446,13983],{"class":517},[507,82448,73558],{"class":576},[507,82450,82451],{"class":517},"()   ",[507,82453,82454],{"class":562},"# tomography in Z, X, Y -> fidelity of the healed butterfly\n",[18,82456,82457],{},"With five qubits and three layers of random all-to-all gates, the field scrambles fast: deeply enough that no single butterfly holds its own state any more. The damage step entangles a throwaway ancilla with one butterfly and discards it, cutting that butterfly off from the field. Then the healing step replays the whole scramble in reverse, and the lost state reassembles from the correlations the others were still holding.",[74616,82459,82461],{"lead":82460},"Quantum Butterfly Field is open source and MIT-licensed, built to be forked and rerun.",[41852,82462,82463,82471],{},[41855,82464,82465],{},[41858,82466,82467,82469],{},[41861,82468,74627],{},[41861,82470,74630],{},[41868,82472,82473,82480,82488,82496],{},[41858,82474,82475,82477],{},[41873,82476,80076],{},[41873,82478,82479],{},"Three.js, custom pipeline, shaders, and flow fields, with a React overlay.",[41858,82481,82482,82485],{},[41873,82483,82484],{},"Motion",[41873,82486,82487],{},"Chaotic attractors and flow fields driving the butterfly movement.",[41858,82489,82490,82493],{},[41873,82491,82492],{},"Quantum simulation",[41873,82494,82495],{},"Python serverless (Vercel) running a Qiskit statevector simulation.",[41858,82497,82498,82501],{},[41873,82499,82500],{},"Quantum hardware",[41873,82502,82503],{},"IonQ Forte via qiskit-ionq, replayed from a recorded-run library.",[13,82505,82507],{"id":82506},"physics-you-feel-not-read","Physics you feel, not read",[18,82509,82510],{},"The discipline of the piece is that every visual property is driven by a real quantum measure, computed layer by layer as the circuit runs. No numbers ever appear on screen. The physics is felt, not read.",[42,82512,82513,82519,82525],{},[45,82514,82515,82518],{},[154,82516,82517],{},"Purity"," drives form: how sharp or translucent each butterfly's wings are, a read on how defined that qubit still is.",[45,82520,82521,82524],{},[154,82522,82523],{},"Quantum mutual information"," drives color mixing and the threads drawn between butterflies, showing what each shares with the others.",[45,82526,82527,82530],{},[154,82528,82529],{},"Fidelity"," drives the luminosity of the damaged butterfly, showing how much of it has returned.",[18,82532,82533,82534,82537,82538,82541],{},"And the data comes from two tracks at once. A ",[154,82535,82536],{},"simulator track"," runs live on every visit: an exact statevector simulation of the full protocol executes on demand in a serverless function, returning per-layer purities, pairwise entanglement, and exact fidelities that drive the animation in real time. A ",[154,82539,82540],{},"hardware track"," is recorded: the damage-and-healing fidelities come from real runs on IonQ Forte, captured offline and replayed from a library, so each visit draws a different recorded run and the healed butterfly's final resting state is anchored in what actually happened on the trapped-ion processor.",[13,82543,82545],{"id":82544},"a-relational-world","A relational world",[18,82547,82548,82549,82552],{},"The default experience is a storyboard in nine beats, an arc from individual identity and interaction through scrambling, rupture, and restoration. The script draws on Federico Faggin's ",[1031,82550,82551],{},"Irreducible"," and resolves into Carlo Rovelli's relational interpretation of quantum mechanics, in which things do not have properties on their own but only in relation to one another.",[72443,82554,82555],{"avatar":104,"name":73820,"role":73821,"username":104},[18,82556,73824],{},[18,82558,82559],{},"That is the throughline that makes the physics feel like more than a demo. The anti-butterfly effect says a part can be lost and still recovered, because it was never only itself. The lōkahi framing says the same thing about people and the relationships they live inside. The circuit is the proof; the butterflies are how you feel it.",[13,82561,73027],{"id":73026},[18,82563,82564],{},"The scrambling circuit is open and forkable on Qollab, the full artwork is MIT-licensed on GitHub, and you can experience the live piece in your browser right now. Change the number of butterflies, the scrambling depth, or which one gets damaged, and rerun it on real hardware.",[73026,82566,82569],{"fork-href":81788,"live-href":82567,"title":82568},"https:\u002F\u002Fquantumbutterflyfield.xyz","Scramble a field, break it, and heal it.",[18,82570,82571,82572],{},"Fork the Quantum Butterfly Field circuit, tune the scrambling, and run the self-healing protocol on real hardware. ",[154,82573,73040],{},[953,82575,77772],{},{"title":104,"searchDepth":105,"depth":105,"links":82577},[82578,82579,82580,82581,82582],{"id":81752,"depth":105,"text":81753},{"id":81774,"depth":105,"text":81775},{"id":82506,"depth":105,"text":82507},{"id":82544,"depth":105,"text":82545},{"id":73026,"depth":105,"text":73027},[112,969,1013],[82585],{"username":1011,"name":73880,"role":82586,"avatar":73934,"bio":82587,"links":82588},"Artist, developer & designer","Xinyi is a multidisciplinary artist and technologist exploring the intersections of nature, spirituality, and computational media. She holds computer-science degrees from MIT and the University of British Columbia, has developed technology for Disney, Pixar, and Google, and won a Best Paper Award at SIGGRAPH MIG for research on generative AI for animation. Her artwork has been exhibited internationally, including at V2_ Lab for the Unstable Media (Rotterdam), Dutch Design Week, the Xarkis Festival, Plexus Projects (New York), and Soft Times Gallery (San Francisco).",[82589,82591,82593,82594],{"label":73068,"href":82590},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fxinyi",{"label":79074,"href":82592},"https:\u002F\u002Fwww.artofxinyi.com\u002Fpagecv",{"label":73059,"href":79204},{"label":979,"href":82595},"https:\u002F\u002Fgithub.com\u002Fxinyiz\u002Fqbf",{"username":8,"name":73075,"role":73076,"avatar":73077},"Xinyi Zhang built an interactive artwork where five butterflies are five qubits. The circuit scrambles their identities into one entangled field, one is damaged, and the quantum anti-butterfly effect heals it.","An interactive artwork where five butterflies are five qubits: scrambled into one entangled field, one is damaged, and the quantum anti-butterfly effect heals it. A Qollab Spring 2026 project.",{"href":81788,"label":82600},"Fork the circuit",{"image":74034,"alt":82602,"liveUrl":82567},"Quantum Butterfly Field: five butterflies as five qubits in an entangled field",{},"\u002Fblog\u002Fquantum-butterfly-field","2026-04-28",[],[82608,82609,82610],{"username":1004,"project":1005,"title":1006,"category":1007,"thumb":1008,"to":1009},{"username":74354,"project":80167,"title":74245,"category":73752,"thumb":80168,"to":74347},{"username":75023,"project":76250,"title":74564,"category":75052,"thumb":76251,"to":74563},{"title":82612,"description":82613},"Quantum Creative Project Showcase: Quantum Butterfly Field","Five butterflies, five qubits. Scrambled into one entangled field, damaged, and healed by the quantum anti-butterfly effect on IonQ.","blog\u002Fquantum-butterfly-field",[73111,143,67672],"EZtjatovFvd7t9KEYpBVzF3QvLdF93c6UmfRa2z-ywc",{"id":82618,"title":82619,"authors":82620,"body":82621,"breadcrumb":83468,"builders":83469,"byline":83478,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":83479,"description":83480,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":83481,"hero":83483,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":83484,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":83485,"publishDate":83486,"readingTime":80935,"related":83487,"relatedProjects":83488,"seo":83492,"stem":83495,"tags":83496,"track":116,"trackName":116,"__hash__":83497},"blog\u002Fblog\u002Fquantum-systemic-oracle.md","Project Showcase: Quantum Systemic Oracle",[74914],{"type":10,"value":82622,"toc":83460},[82623,82626,82633,82636,82641,82645,82648,82653,82656,82660,82663,82668,82671,82691,82694,82698,82705,82709,82712,83359,83362,83365,83411,83415,83418,83423,83426,83428,83431,83436,83439,83441,83444,83449,83458],[18,82624,82625],{},"The Quantum Systemic Oracle takes a quantum computation and turns it into something a smart contract can read: a single daily number for how stressed the financial system looks, published on-chain for anyone to consume.",[18,82627,82628,82629,82632],{},"Its framing, in the project's own words, is ",[1031,82630,82631],{},"quantum compute as an on-chain primitive",". A portfolio-optimization circuit runs on IonQ, its result is distilled into a systemic risk index in basis points, and that index is pushed to a Chainlink-shaped oracle that other contracts can call like any other price feed.",[18,82634,82635],{},"It is a Spring 2026 challenge project from Jamie Dominguez, who does not come from quantum at all. He spent more than a decade in enterprise data governance at a global bank, and used the challenge to find out how far a domain expert with modern AI tooling could actually get on real quantum hardware.",[72443,82637,82638],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,82639,82640],{},"I came from technology and data governance. I became interested in quantum having long followed tech trends, including using GPUs to speed up data queries, and I believe we'll see similar advancements with quantum technology.",[13,82642,82644],{"id":82643},"from-data-governance-to-a-qpu","From data governance to a QPU",[18,82646,82647],{},"Jamie has spent his career close to large financial data, the reference tables and integrations that keep a bank's systems agreeing with each other. Quantum was a trend he watched from that vantage point, in the same way he had watched GPUs go from a graphics curiosity to the backbone of fast data queries. The challenge was the first time he ran his own jobs rather than reading about other people's.",[72443,82649,82650],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,82651,82652],{},"This was my first QPU job and Qiskit run. I learned it's much easier to get started now than ever before.",[18,82654,82655],{},"That starting point shapes the whole project. It is not a research-grade quantum-finance paper, and it does not pretend to be. It is a working end-to-end pipeline built by someone who knows financial systems deeply and treated the quantum step as one component to wire in, not a mountain to summit first.",[13,82657,82659],{"id":82658},"a-risk-score-as-a-primitive","A risk score as a primitive",[18,82661,82662],{},"The core idea is to make a quantum computation legible to the rest of the software world. Most quantum results live in notebooks. Jamie wanted his to live somewhere other programs could act on it automatically, so he published it where automated systems already read their inputs: on-chain.",[72443,82664,82665],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,82666,82667],{},"The idea of a quantum risk score being on-chain came from thinking about how quantum compute could be leveraged in a wider ecosystem. Blockchain makes sense as a way to potentially contribute to agentic smart contracts in the future, blending quantum, AI, and blockchain.",[18,82669,82670],{},"Concretely, the system is three layers stacked end to end:",[42,82672,82673,82679,82685],{},[45,82674,82675,82678],{},[154,82676,82677],{},"A Python engine"," pulls live market signals, prediction-market odds, funding rates, volatility, and fear indices, and turns them into the inputs for an optimization problem.",[45,82680,82681,82684],{},[154,82682,82683],{},"A quantum step"," runs that optimization on IonQ and blends the result with eight market signals into a single systemic risk index, scaled in basis points from 0 to 10,000.",[45,82686,82687,82690],{},[154,82688,82689],{},"An on-chain oracle"," publishes the index to Ethereum through a Chainlink-shaped AggregatorV3 interface, so any contract can read it with the same call it would use for a price feed.",[18,82692,82693],{},"Once it is on-chain, the score stops being a chart and becomes a building block. A lending protocol could widen collateral ratios when the index spikes, a vault could trigger deleveraging, a prediction market could resolve against it. That is what Jamie means by a primitive: not a dashboard people look at, but a number other code is built on.",[13,82695,82697],{"id":82696},"inside-the-circuit","Inside the circuit",[18,82699,82700,82701,82704],{},"The quantum step is a ",[154,82702,82703],{},"QAOA"," portfolio optimization, the same family of algorithm you would reach for to pick a basket of assets under competing constraints. Jamie chose finance deliberately, because it is one of the areas where quantum methods are already showing early promise.",[72443,82706,82707],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,82708,74917],{},[18,82710,82711],{},"The circuit encodes eight candidate assets as eight decision qubits, plus six more that carry the global market state, the regime, volatility stress, DeFi stress, and stablecoin dominance. A neat piece of engineering keeps it lean: those market witnesses fold into single-qubit rotations rather than expensive two-qubit gates, so the circuit grows by exactly one qubit per asset, not per signal. On Qollab, the circuit-side preview runs on hardware exactly as written:",[493,82713,82716],{"name":82714,"run-href":82715,"tag":72511},"Qollab.py","\u002Fu\u002Fjamie\u002Fquantum-systemic-oracle",[498,82717,82719],{"className":500,"code":82718,"language":502,"meta":72515,"style":104},"# The quantum step of a daily on-chain crypto risk oracle.\n# 'backend' is pre-created from the \"Select QPU\" dropdown below.\nfrom qiskit import QuantumCircuit, transpile\nfrom qiskit.providers.jobstatus import JobStatus\nimport time\n\nN_ASSETS, N_WITNESS, SHOTS = 8, 6, 1000\n\ndef apply_cost_layer(qc, gamma, h, J_pairs, n):\n    # ZZ couplings = the portfolio cost Hamiltonian\n    for i in range(n):\n        qc.rz(2.0 * gamma * h[i], i)\n    for p in J_pairs:\n        qc.cx(p[\"i\"], p[\"j\"])\n        qc.rz(2.0 * gamma * p[\"value\"], p[\"j\"])\n        qc.cx(p[\"i\"], p[\"j\"])\n\ndef build_circuit(gammas, betas, h, J_pairs, snapshot):\n    n, n_total = N_ASSETS, N_ASSETS + N_WITNESS   # 8 assets + 6 witnesses\n    qc = QuantumCircuit(n_total, n_total)\n    for i in range(n):\n        qc.h(i)\n\n    stressed = snapshot[\"regime\"][\"regime\"] == \"risk_off\"\n    def fold(theta, p):                       # witness -> asset soft bias\n        for i in range(n):\n            qc.ry(theta * (1.0 - 2.0 * p) \u002F 2.0, i)\n    fold(-0.10, 1.0 if stressed else 0.0)\n\n    for layer in range(len(gammas)):\n        apply_cost_layer(qc, gammas[layer], h, J_pairs, n)\n        for i in range(n):\n            qc.rx(2.0 * betas[layer], i)\n    qc.measure(range(n_total), range(n_total))\n    return qc\n\nqc = build_circuit(gammas, betas, ising_h, ising_J_pairs, SNAPSHOT)\njob = backend.run(transpile(qc, backend), shots=SHOTS)\nwhile job.status() is not JobStatus.DONE:\n    time.sleep(2)\ncounts = job.result().get_counts()   # sampled portfolios -> risk score (BPS)\n",[504,82720,82721,82726,82730,82740,82750,82756,82760,82787,82791,82823,82828,82840,82860,82871,82891,82927,82943,82947,82978,83000,83011,83023,83039,83043,83067,83090,83102,83133,83165,83169,83187,83195,83207,83229,83247,83253,83257,83273,83308,83328,83340],{"__ignoreMap":104},[507,82722,82723],{"class":509,"line":510},[507,82724,82725],{"class":562},"# The quantum step of a daily on-chain crypto risk oracle.\n",[507,82727,82728],{"class":509,"line":105},[507,82729,79555],{"class":562},[507,82731,82732,82734,82736,82738],{"class":509,"line":540},[507,82733,529],{"class":513},[507,82735,532],{"class":517},[507,82737,514],{"class":513},[507,82739,77077],{"class":517},[507,82741,82742,82744,82746,82748],{"class":509,"line":553},[507,82743,529],{"class":513},[507,82745,73222],{"class":517},[507,82747,514],{"class":513},[507,82749,73227],{"class":517},[507,82751,82752,82754],{"class":509,"line":559},[507,82753,514],{"class":513},[507,82755,80323],{"class":517},[507,82757,82758],{"class":509,"line":566},[507,82759,556],{"emptyLinePlaceholder":133},[507,82761,82762,82765,82767,82770,82772,82774,82776,82778,82780,82782,82784],{"class":509,"line":590},[507,82763,82764],{"class":583},"N_ASSETS",[507,82766,622],{"class":517},[507,82768,82769],{"class":583},"N_WITNESS",[507,82771,622],{"class":517},[507,82773,73263],{"class":583},[507,82775,1423],{"class":572},[507,82777,64732],{"class":583},[507,82779,622],{"class":517},[507,82781,63712],{"class":583},[507,82783,622],{"class":517},[507,82785,82786],{"class":583},"1000\n",[507,82788,82789],{"class":509,"line":610},[507,82790,556],{"emptyLinePlaceholder":133},[507,82792,82793,82795,82798,82800,82803,82805,82808,82810,82812,82814,82817,82819,82821],{"class":509,"line":634},[507,82794,1370],{"class":513},[507,82796,82797],{"class":576}," apply_cost_layer",[507,82799,580],{"class":517},[507,82801,82802],{"class":1382},"qc",[507,82804,622],{"class":517},[507,82806,82807],{"class":1382},"gamma",[507,82809,622],{"class":517},[507,82811,596],{"class":1382},[507,82813,622],{"class":517},[507,82815,82816],{"class":1382},"J_pairs",[507,82818,622],{"class":517},[507,82820,4420],{"class":1382},[507,82822,1883],{"class":517},[507,82824,82825],{"class":509,"line":661},[507,82826,82827],{"class":562},"    # ZZ couplings = the portfolio cost Hamiltonian\n",[507,82829,82830,82832,82834,82836,82838],{"class":509,"line":678},[507,82831,1916],{"class":513},[507,82833,8246],{"class":517},[507,82835,1636],{"class":513},[507,82837,8221],{"class":572},[507,82839,76436],{"class":517},[507,82841,82842,82844,82846,82848,82850,82852,82855,82857],{"class":509,"line":683},[507,82843,72955],{"class":517},[507,82845,80508],{"class":576},[507,82847,580],{"class":517},[507,82849,59542],{"class":583},[507,82851,8229],{"class":572},[507,82853,82854],{"class":517}," gamma ",[507,82856,2391],{"class":572},[507,82858,82859],{"class":517}," h[i], i)\n",[507,82861,82862,82864,82866,82868],{"class":509,"line":697},[507,82863,1916],{"class":513},[507,82865,68242],{"class":517},[507,82867,1636],{"class":513},[507,82869,82870],{"class":517}," J_pairs:\n",[507,82872,82873,82875,82877,82880,82883,82886,82889],{"class":509,"line":710},[507,82874,72955],{"class":517},[507,82876,615],{"class":576},[507,82878,82879],{"class":517},"(p[",[507,82881,82882],{"class":730},"\"i\"",[507,82884,82885],{"class":517},"], p[",[507,82887,82888],{"class":730},"\"j\"",[507,82890,68725],{"class":517},[507,82892,82893,82895,82897,82899,82901,82903,82905,82907,82910,82913,82915,82917,82919],{"class":509,"line":715},[507,82894,72955],{"class":517},[507,82896,80508],{"class":576},[507,82898,580],{"class":517},[507,82900,59542],{"class":583},[507,82902,8229],{"class":572},[507,82904,82854],{"class":517},[507,82906,2391],{"class":572},[507,82908,82909],{"class":517}," p[",[507,82911,82912],{"class":730},"\"value\"",[507,82914,82885],{"class":517},[507,82916,82888],{"class":730},[507,82918,79797],{"class":517},[507,82920,72708,82921],{"class":72706,"tabindex":72707},[507,82922,82923,82926],{"class":72711,"role":72712},[154,82924,82925],{},"Cost layer."," Each ZZ coupling encodes a pairwise term of the portfolio Hamiltonian: Markowitz risk, funding crowding, and prediction-market sentiment.",[507,82928,82929,82931,82933,82935,82937,82939,82941],{"class":509,"line":721},[507,82930,72955],{"class":517},[507,82932,615],{"class":576},[507,82934,82879],{"class":517},[507,82936,82882],{"class":730},[507,82938,82885],{"class":517},[507,82940,82888],{"class":730},[507,82942,68725],{"class":517},[507,82944,82945],{"class":509,"line":736},[507,82946,556],{"emptyLinePlaceholder":133},[507,82948,82949,82951,82953,82955,82958,82960,82963,82965,82967,82969,82971,82973,82976],{"class":509,"line":748},[507,82950,1370],{"class":513},[507,82952,72812],{"class":576},[507,82954,580],{"class":517},[507,82956,82957],{"class":1382},"gammas",[507,82959,622],{"class":517},[507,82961,82962],{"class":1382},"betas",[507,82964,622],{"class":517},[507,82966,596],{"class":1382},[507,82968,622],{"class":517},[507,82970,82816],{"class":1382},[507,82972,622],{"class":517},[507,82974,82975],{"class":1382},"snapshot",[507,82977,1883],{"class":517},[507,82979,82980,82983,82985,82988,82990,82992,82994,82997],{"class":509,"line":761},[507,82981,82982],{"class":517},"    n, n_total ",[507,82984,573],{"class":572},[507,82986,82987],{"class":583}," N_ASSETS",[507,82989,622],{"class":517},[507,82991,82764],{"class":583},[507,82993,8313],{"class":572},[507,82995,82996],{"class":583}," N_WITNESS",[507,82998,82999],{"class":562},"   # 8 assets + 6 witnesses\n",[507,83001,83002,83004,83006,83008],{"class":509,"line":775},[507,83003,72833],{"class":517},[507,83005,573],{"class":572},[507,83007,577],{"class":576},[507,83009,83010],{"class":517},"(n_total, n_total)\n",[507,83012,83013,83015,83017,83019,83021],{"class":509,"line":784},[507,83014,1916],{"class":513},[507,83016,8246],{"class":517},[507,83018,1636],{"class":513},[507,83020,8221],{"class":572},[507,83022,76436],{"class":517},[507,83024,83025,83027,83029,83032],{"class":509,"line":796},[507,83026,72955],{"class":517},[507,83028,596],{"class":576},[507,83030,83031],{"class":517},"(i)",[507,83033,72708,83034],{"class":72706,"tabindex":72707},[507,83035,83036,83038],{"class":72711,"role":72712},[154,83037,73373],{}," A Hadamard on each asset qubit opens an even superposition over all 256 candidate portfolios: QAOA's starting point.",[507,83040,83041],{"class":509,"line":809},[507,83042,556],{"emptyLinePlaceholder":133},[507,83044,83045,83048,83050,83053,83056,83058,83060,83062,83064],{"class":509,"line":1352},[507,83046,83047],{"class":517},"    stressed ",[507,83049,573],{"class":572},[507,83051,83052],{"class":517}," snapshot[",[507,83054,83055],{"class":730},"\"regime\"",[507,83057,1755],{"class":517},[507,83059,83055],{"class":730},[507,83061,8206],{"class":517},[507,83063,1723],{"class":572},[507,83065,83066],{"class":730}," \"risk_off\"\n",[507,83068,83069,83072,83075,83077,83080,83082,83084,83087],{"class":509,"line":1357},[507,83070,83071],{"class":513},"    def",[507,83073,83074],{"class":576}," fold",[507,83076,580],{"class":517},[507,83078,83079],{"class":1382},"theta",[507,83081,622],{"class":517},[507,83083,18],{"class":1382},[507,83085,83086],{"class":517},"):                       ",[507,83088,83089],{"class":562},"# witness -> asset soft bias\n",[507,83091,83092,83094,83096,83098,83100],{"class":509,"line":1362},[507,83093,2267],{"class":513},[507,83095,8246],{"class":517},[507,83097,1636],{"class":513},[507,83099,8221],{"class":572},[507,83101,76436],{"class":517},[507,83103,83104,83106,83108,83111,83113,83115,83117,83119,83121,83123,83126,83128,83130],{"class":509,"line":1367},[507,83105,72891],{"class":517},[507,83107,639],{"class":576},[507,83109,83110],{"class":517},"(theta ",[507,83112,2391],{"class":572},[507,83114,58644],{"class":517},[507,83116,57927],{"class":583},[507,83118,65914],{"class":572},[507,83120,57579],{"class":583},[507,83122,8229],{"class":572},[507,83124,83125],{"class":517}," p) ",[507,83127,645],{"class":572},[507,83129,57579],{"class":583},[507,83131,83132],{"class":517},", i)\n",[507,83134,83135,83138,83140,83142,83145,83147,83149,83151,83154,83156,83158,83160],{"class":509,"line":1379},[507,83136,83137],{"class":576},"    fold",[507,83139,580],{"class":517},[507,83141,2367],{"class":572},[507,83143,83144],{"class":583},"0.10",[507,83146,622],{"class":517},[507,83148,57927],{"class":583},[507,83150,66162],{"class":513},[507,83152,83153],{"class":517}," stressed ",[507,83155,61407],{"class":513},[507,83157,57367],{"class":583},[507,83159,3649],{"class":517},[507,83161,72708,83162],{"class":72706,"tabindex":72707},[507,83163,83164],{"class":72711,"role":72712},"Six global market witnesses (regime, vol-stress, DeFi-stress, stablecoin dominance) fold into 1-qubit ry rotations, so a risk-off market tilts every asset defensive without extra two-qubit gates.",[507,83166,83167],{"class":509,"line":1389},[507,83168,556],{"emptyLinePlaceholder":133},[507,83170,83171,83173,83176,83178,83180,83182,83184],{"class":509,"line":1397},[507,83172,1916],{"class":513},[507,83174,83175],{"class":517}," layer ",[507,83177,1636],{"class":513},[507,83179,8221],{"class":572},[507,83181,580],{"class":517},[507,83183,1763],{"class":572},[507,83185,83186],{"class":517},"(gammas)):\n",[507,83188,83189,83192],{"class":509,"line":1412},[507,83190,83191],{"class":576},"        apply_cost_layer",[507,83193,83194],{"class":517},"(qc, gammas[layer], h, J_pairs, n)\n",[507,83196,83197,83199,83201,83203,83205],{"class":509,"line":1431},[507,83198,2267],{"class":513},[507,83200,8246],{"class":517},[507,83202,1636],{"class":513},[507,83204,8221],{"class":572},[507,83206,76436],{"class":517},[507,83208,83209,83211,83213,83215,83217,83219,83222],{"class":509,"line":1449},[507,83210,72891],{"class":517},[507,83212,80553],{"class":576},[507,83214,580],{"class":517},[507,83216,59542],{"class":583},[507,83218,8229],{"class":572},[507,83220,83221],{"class":517}," betas[layer], i)",[507,83223,72708,83224],{"class":72706,"tabindex":72707},[507,83225,83226,83228],{"class":72711,"role":72712},[154,83227,80569],{}," The RX layer nudges the state toward neighbouring portfolios so the optimizer can escape a single bitstring.",[507,83230,83231,83233,83235,83237,83239,83242,83244],{"class":509,"line":1465},[507,83232,21867],{"class":517},[507,83234,72822],{"class":576},[507,83236,580],{"class":517},[507,83238,2204],{"class":572},[507,83240,83241],{"class":517},"(n_total), ",[507,83243,2204],{"class":572},[507,83245,83246],{"class":517},"(n_total))\n",[507,83248,83249,83251],{"class":509,"line":1471},[507,83250,2504],{"class":513},[507,83252,72990],{"class":517},[507,83254,83255],{"class":509,"line":1477},[507,83256,556],{"emptyLinePlaceholder":133},[507,83258,83259,83261,83263,83265,83268,83271],{"class":509,"line":1482},[507,83260,569],{"class":517},[507,83262,573],{"class":572},[507,83264,72812],{"class":576},[507,83266,83267],{"class":517},"(gammas, betas, ising_h, ising_J_pairs, ",[507,83269,83270],{"class":583},"SNAPSHOT",[507,83272,587],{"class":517},[507,83274,83275,83277,83279,83281,83283,83285,83288,83291,83293,83295,83297,83299],{"class":509,"line":1488},[507,83276,23964],{"class":517},[507,83278,573],{"class":572},[507,83280,73487],{"class":517},[507,83282,22501],{"class":576},[507,83284,580],{"class":517},[507,83286,83287],{"class":576},"transpile",[507,83289,83290],{"class":517},"(qc, backend), ",[507,83292,68762],{"class":2155},[507,83294,573],{"class":572},[507,83296,73263],{"class":583},[507,83298,3649],{"class":517},[507,83300,72708,83301],{"class":72706,"tabindex":72707},[507,83302,83303,83304,83307],{"class":72711,"role":72712},"Submits one shot batch to the selected IonQ backend through Qollab. The sampled portfolios become this run's ",[504,83305,83306],{},"qaoa_solution_quality"," component of the risk index.",[507,83309,83310,83312,83314,83316,83318,83320,83322,83324,83326],{"class":509,"line":1494},[507,83311,79882],{"class":513},[507,83313,23993],{"class":517},[507,83315,73518],{"class":576},[507,83317,1677],{"class":517},[507,83319,37008],{"class":513},[507,83321,21980],{"class":513},[507,83323,73527],{"class":517},[507,83325,73530],{"class":583},[507,83327,1728],{"class":517},[507,83329,83330,83332,83334,83336,83338],{"class":509,"line":1500},[507,83331,79903],{"class":517},[507,83333,73540],{"class":576},[507,83335,580],{"class":517},[507,83337,584],{"class":583},[507,83339,587],{"class":517},[507,83341,83342,83344,83346,83348,83350,83352,83354,83356],{"class":509,"line":1506},[507,83343,79916],{"class":517},[507,83345,573],{"class":572},[507,83347,23993],{"class":517},[507,83349,23996],{"class":576},[507,83351,13983],{"class":517},[507,83353,73558],{"class":576},[507,83355,82451],{"class":517},[507,83357,83358],{"class":562},"# sampled portfolios -> risk score (BPS)\n",[18,83360,83361],{},"The snapshot baked into the gallery version is frozen on a single day so it runs without any network calls, but the full pipeline in the repo fetches fresh market data daily and reruns the whole chain.",[831,83363],{"caption":83364,"no":835,"poster":74907,"video":74908},"A daily run end to end: live market data in, a QAOA job on IonQ, the systemic risk index in basis points, and the on-chain publication preview. Press play.",[74616,83366,83368],{"lead":83367},"The Quantum Systemic Oracle is open source, built to be forked and rerun.",[41852,83369,83370,83378],{},[41855,83371,83372],{},[41858,83373,83374,83376],{},[41861,83375,74627],{},[41861,83377,74630],{},[41868,83379,83380,83387,83395,83403],{},[41858,83381,83382,83384],{},[41873,83383,72215],{},[41873,83385,83386],{},"Qiskit + IonQ SDK, a 14-qubit QAOA on IonQ (Forte-class).",[41858,83388,83389,83392],{},[41873,83390,83391],{},"Engine",[41873,83393,83394],{},"Python 3.11+, live market ingest, QUBO build, and risk scoring.",[41858,83396,83397,83400],{},[41873,83398,83399],{},"Oracle",[41873,83401,83402],{},"Solidity + Chainlink AggregatorV3, on Ethereum Sepolia.",[41858,83404,83405,83408],{},[41873,83406,83407],{},"Built with",[41873,83409,83410],{},"Cursor Agent and Chainlink Agent Skills (AI pair-programming).",[13,83412,83414],{"id":83413},"built-with-ai-on-real-hardware","Built with AI, on real hardware",[18,83416,83417],{},"Jamie is candid that he did not write every line of Qiskit and Solidity from memory. He treated modern AI coding tools as the thing that closed the gap between his domain knowledge and the unfamiliar quantum and blockchain stacks, and he is enthusiastic about it as a way in.",[72443,83419,83420],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,83421,83422],{},"It was enjoyable leveraging Cursor and the latest LLMs to create what was made. I'd encourage everyone to do so, it's the best way to learn first hand.",[18,83424,83425],{},"His practical advice for anyone wanting to follow the same path is concrete: set up MCP servers and lean on the basic skills they expose to start building straight away. The point is not to outsource the understanding, but to get a working loop going fast enough that you actually learn by running things, which is exactly how he got from never having touched Qiskit to submitting jobs on IonQ.",[13,83427,73675],{"id":73674},[18,83429,83430],{},"The current oracle is a deliberately bounded prototype: eight assets, a frozen snapshot in the gallery, and publication confined to a testnet. What Jamie is watching is the hardware curve, because the use cases he is interested in open up as the machines grow.",[72443,83432,83433],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,83434,83435],{},"I enjoyed running my first QPU jobs. I'll continue my learning path and look forward to running additional experiments as logical qubits scale, since that will keep enabling new use cases.",[18,83437,83438],{},"The architecture is built to grow with that curve. Because each asset adds exactly one qubit, widening the basket is a matter of appending to the asset universe and letting the circuit width, budget, and readout adapt. The longer arc he points at, agentic smart contracts that blend quantum, AI, and blockchain, is speculative by his own admission, but the oracle is a small, working first step in that direction.",[13,83440,73027],{"id":73026},[18,83442,83443],{},"The whole pipeline is open and forkable, from the circuit-side preview on Qollab to the live-data engine and the on-chain publisher on GitHub. If you want to go all the way to publishing your own score, Jamie has a tip for getting unstuck on the blockchain side.",[72443,83445,83446],{"avatar":74911,"name":74912,"role":74913,"username":74914},[18,83447,83448],{},"I'd suggest others try getting testnet Chainlink and ETH tokens to test and publish their own smart contracts.",[73026,83450,83453],{"fork-href":82715,"live-href":83451,"title":83452},"https:\u002F\u002Fkedwind.github.io\u002FQuantum-Systemic-Oracle\u002F","Turn a quantum result into something code can read.",[18,83454,83455,83456],{},"Fork the Quantum Systemic Oracle, run the QAOA step on real hardware, and publish your own score on-chain. ",[154,83457,73040],{},[953,83459,77772],{},{"title":104,"searchDepth":105,"depth":105,"links":83461},[83462,83463,83464,83465,83466,83467],{"id":82643,"depth":105,"text":82644},{"id":82658,"depth":105,"text":82659},{"id":82696,"depth":105,"text":82697},{"id":83413,"depth":105,"text":83414},{"id":73674,"depth":105,"text":73675},{"id":73026,"depth":105,"text":73027},[112,969,74523],[83470],{"username":74914,"name":74912,"role":83471,"avatar":74911,"bio":83472,"links":83473},"Lead architect & developer","Jamie is a data-governance and finance professional with more than a decade managing enterprise data architecture at a global bank, currently an Apps Dev Group Manager working across reference data, metadata curation, and system integrations for large financial platforms. He describes himself as a domain expert rather than a traditional quantum engineer, and built the Quantum Systemic Oracle as a solo project to see how far AI-assisted development could take him on real hardware. It was his first QPU job and first Qiskit run.",[83474,83476],{"label":73068,"href":83475},"https:\u002F\u002Fqollab.xyz\u002Fu\u002Fjamie",{"label":979,"href":83477},"https:\u002F\u002Fgithub.com\u002Fkedwind",{"username":8,"name":73075,"role":73076,"avatar":73077},"Jamie Dominguez built a daily oracle that runs a portfolio-optimization circuit on IonQ, distills the result into a single systemic risk score, and publishes it on-chain for smart contracts to read.","Jamie Dominguez built a daily oracle: a portfolio-optimization circuit on IonQ distilled into one systemic risk score, published on-chain. A Qollab project.",{"href":82715,"label":83482},"Fork the oracle",{"image":104,"alt":104,"liveUrl":83451},{},"\u002Fblog\u002Fquantum-systemic-oracle","2026-04-26",[],[83489,83490,83491],{"username":75023,"project":76250,"title":74564,"category":75052,"thumb":76251,"to":74563},{"username":997,"project":998,"title":999,"category":73752,"thumb":1001,"to":1002},{"username":74390,"project":77806,"title":74277,"category":73752,"thumb":77807,"to":74383},{"title":83493,"description":83494},"Quantum Creative Project Showcase: Quantum Systemic Oracle","A daily systemic risk score, computed by a QAOA circuit on IonQ and published on-chain as a primitive smart contracts can read.","blog\u002Fquantum-systemic-oracle",[75067,143,75068],"oCHhjTCuKEhOIU6gDjDqqdZYENF6EGMSz0_4AP3S5JA",{"id":83499,"title":83500,"authors":83501,"body":83502,"breadcrumb":84120,"builders":84121,"byline":84131,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":84132,"description":84133,"draft":125,"extension":126,"eyebrow":73080,"finish":116,"fork":84134,"hero":84136,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":84138,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":84139,"publishDate":84140,"readingTime":993,"related":84141,"relatedProjects":84142,"seo":84146,"stem":84149,"tags":84150,"track":116,"trackName":116,"__hash__":84152},"blog\u002Fblog\u002Fquantum-canvas.md","Project Showcase: QuantumCanvas",[74336],{"type":10,"value":83503,"toc":84113},[83504,83507,83514,83517,83520,83525,83529,83532,83537,83540,83560,83563,83567,83570,83573,83599,83602,83607,83610,83615,83626,83988,84033,84037,84040,84045,84048,84050,84053,84058,84061,84081,84092,84094,84097,84102,84111],[18,83505,83506],{},"QuantumCanvas is a visual sandbox where you drag and drop quantum operations to build new algorithms, then run them on real hardware.",[18,83508,83509,83510,83513],{},"Its guiding line, in the project's words, is ",[1031,83511,83512],{},"thinking in logic, computing in quantum",", the idea that curiosity should be all it takes to start exploring, no background in linear algebra required.",[18,83515,83516],{},"It is built for a specific moment: you have finished the tutorials, you understand the theory at a high level, and then you hit a wall. Building a new circuit from scratch feels intimidating and repetitive. QuantumCanvas is the bridge across that gap, a place to experiment, observe patterns, and develop intuition without getting stuck behind heavy math or code.",[18,83518,83519],{},"It is a featured project from Qollab's Creative Challenge, built by Shivani Mayekar, a Georgia Tech computer-science researcher who has taught quantum to hundreds of people and kept hearing the same problem.",[72443,83521,83523],{"avatar":83522,"name":74240,"role":74335,"username":74336},"\u002F_content\u002Fimages\u002Fbuilders\u002Fshivani-mayekar.webp",[18,83524,74339],{},[13,83526,83528],{"id":83527},"the-gap-after-the-tutorials","The gap after the tutorials",[18,83530,83531],{},"Shivani has spent years on the teaching side of quantum. She co-founded Qtangled and has run workshops for more than 250 people, and across all those rooms one problem kept surfacing. Learners could follow the theory at a high level, but they had almost nowhere to go next.",[72443,83533,83534],{"avatar":83522,"name":74240,"role":74335,"username":74336},[18,83535,83536],{},"The same challenge came up repeatedly: how do people experiment, discover, and build intuition in quantum computing without getting stuck behind complex mathematics or code? Most learners could understand the theory at a high level, but they had very few opportunities to explore concepts in an intuitive way. QuantumCanvas grew from a simple question: how can we make experimentation and discovery easier while still teaching the underlying logic of quantum systems?",[18,83538,83539],{},"QuantumCanvas treats this as a human-computer-interaction problem as much as a physics one. It names three walls every newcomer runs into, and sets out to lower each:",[42,83541,83542,83548,83554],{},[45,83543,83544,83547],{},[154,83545,83546],{},"The math wall",": vector spaces and unitary matrices.",[45,83549,83550,83553],{},[154,83551,83552],{},"The physics wall",": hardware noise, decoherence, and gate timing.",[45,83555,83556,83559],{},[154,83557,83558],{},"The syntax wall",": learning a low-level framework before you see a single result.",[18,83561,83562],{},"That question is the whole design brief. Not another tutorial, and not a research-grade toolchain, but a place where the post-tutorial learner can actually build something and watch what it does.",[13,83564,83566],{"id":83565},"shake-mark-boost-link","Shake, mark, boost, link",[18,83568,83569],{},"QuantumCanvas is a grid you compose on. Instead of writing code, you drag and drop a small set of plain-language operations and arrange them on the canvas, with the tool showing you both the circuit and the live state visualization side by side as you go. When you want the real thing, you run it on hardware.",[18,83571,83572],{},"Each tile maps to a genuine quantum action, and the canvas enforces the order they have to happen in:",[42,83574,83575,83581,83587,83593],{},[45,83576,83577,83580],{},[154,83578,83579],{},"Shake"," spreads every possibility equally, the opening move from a ground state.",[45,83582,83583,83586],{},[154,83584,83585],{},"Mark"," tags a target with a hidden signal, which only works once you have shaken into superposition.",[45,83588,83589,83592],{},[154,83590,83591],{},"Boost"," amplifies the marked item so it becomes the one you are most likely to measure.",[45,83594,83595,83598],{},[154,83596,83597],{},"Link"," entangles two qubits so they move together.",[18,83600,83601],{},"Shivani explored several ways to visualize quantum logic before settling on this grid. The point was to make the underlying ideas tangible without watering them down.",[72443,83603,83604],{"avatar":83522,"name":74240,"role":74335,"username":74336},[18,83605,83606],{},"I wanted something that felt intuitive without losing the essence of the underlying concepts. The grid system allows people to experiment, observe patterns, and develop intuition through interaction. It stays grounded in real quantum principles while making the learning process more approachable and engaging.",[18,83608,83609],{},"Under the hood, those operations are a modular library of quantum primitives, reusable building blocks you can recombine into new algorithms. On Qollab, that library lives in the Playground as a forkable collection, so the canvas is not just a teaching demo but a starting point other people can build on.",[831,83611],{"caption":83612,"no":835,"poster":83613,"video":83614},"Composing on the canvas: drop qubits, shake them into superposition, mark and boost a target, and link qubits, with the circuit and live state updating as you go. Press play.","\u002F_content\u002Fimages\u002Fquantum-canvas\u002Fhero.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002Fa358575f-01fc-46d7-aa1d-29e4f89fd295",[18,83616,83617,83618,83621,83622,83625],{},"It is all real code underneath. A bare Bell state, for instance, is just ",[154,83619,83620],{},"shake"," then ",[154,83623,83624],{},"link",", and on Qollab the Playground runs it on hardware exactly as written:",[493,83627,83629],{"name":81172,"run-href":83628,"tag":496},"\u002Fu\u002FShivaniMayekar\u002Fquantum-canvas",[498,83630,83632],{"className":500,"code":83631,"language":502,"meta":104,"style":104},"# 'backend' is pre-created as a global on Qollab\nfrom qiskit import QuantumCircuit\n\n# A Bell state: shake (h) then link (cx)\ncircuit = QuantumCircuit(2, 2)\ncircuit.h(0)\ncircuit.cx(0, 1)\ncircuit.measure([0, 1], [0, 1])\nprint(circuit)\n\nfrom qiskit.providers.jobstatus import JobStatus\nimport time\n\ndef main(shots=100, low_prob=0.05):\n    job = backend.run(circuit, shots=shots)\n\n    # Poll until the job finishes\n    while True:\n        status = job.status()\n        print(f\"Job status is {status}\")\n        if status is JobStatus.DONE:\n            break\n        time.sleep(10)\n\n    counts = job.get_counts()\n    # Filter low-probability noise\n    counts = {b: c for b, c in counts.items() if c > shots * low_prob}\n    print(f\"Counts for {shots} shots: {counts}\")\n",[504,83633,83634,83638,83648,83652,83657,83675,83695,83719,83743,83749,83753,83763,83769,83773,83798,83823,83827,83831,83839,83851,83871,83885,83889,83901,83905,83917,83922,83960],{"__ignoreMap":104},[507,83635,83636],{"class":509,"line":510},[507,83637,73205],{"class":562},[507,83639,83640,83642,83644,83646],{"class":509,"line":105},[507,83641,529],{"class":513},[507,83643,532],{"class":517},[507,83645,514],{"class":513},[507,83647,537],{"class":517},[507,83649,83650],{"class":509,"line":540},[507,83651,556],{"emptyLinePlaceholder":133},[507,83653,83654],{"class":509,"line":553},[507,83655,83656],{"class":562},"# A Bell state: shake (h) then link (cx)\n",[507,83658,83659,83661,83663,83665,83667,83669,83671,83673],{"class":509,"line":559},[507,83660,73339],{"class":517},[507,83662,573],{"class":572},[507,83664,577],{"class":576},[507,83666,580],{"class":517},[507,83668,584],{"class":583},[507,83670,622],{"class":517},[507,83672,584],{"class":583},[507,83674,587],{"class":517},[507,83676,83677,83679,83681,83683,83685,83687],{"class":509,"line":566},[507,83678,73358],{"class":517},[507,83680,596],{"class":576},[507,83682,580],{"class":517},[507,83684,601],{"class":583},[507,83686,3649],{"class":517},[507,83688,72708,83689],{"class":72706,"tabindex":72707},[507,83690,83691,83694],{"class":72711,"role":72712},[154,83692,83693],{},"Shake."," A Hadamard gate puts the qubit into an even superposition of 0 and 1.",[507,83696,83697,83699,83701,83703,83705,83707,83709,83711],{"class":509,"line":590},[507,83698,73358],{"class":517},[507,83700,615],{"class":576},[507,83702,580],{"class":517},[507,83704,601],{"class":583},[507,83706,622],{"class":517},[507,83708,625],{"class":583},[507,83710,3649],{"class":517},[507,83712,72708,83713],{"class":72706,"tabindex":72707},[507,83714,83715,83718],{"class":72711,"role":72712},[154,83716,83717],{},"Link."," A CNOT entangles qubit 1 with qubit 0, so measuring one tells you about the other.",[507,83720,83721,83723,83725,83727,83729,83731,83733,83735,83737,83739,83741],{"class":509,"line":610},[507,83722,73358],{"class":517},[507,83724,72822],{"class":576},[507,83726,79780],{"class":517},[507,83728,601],{"class":583},[507,83730,622],{"class":517},[507,83732,625],{"class":583},[507,83734,75921],{"class":517},[507,83736,601],{"class":583},[507,83738,622],{"class":517},[507,83740,625],{"class":583},[507,83742,68725],{"class":517},[507,83744,83745,83747],{"class":509,"line":634},[507,83746,8525],{"class":572},[507,83748,81296],{"class":517},[507,83750,83751],{"class":509,"line":661},[507,83752,556],{"emptyLinePlaceholder":133},[507,83754,83755,83757,83759,83761],{"class":509,"line":678},[507,83756,529],{"class":513},[507,83758,73222],{"class":517},[507,83760,514],{"class":513},[507,83762,73227],{"class":517},[507,83764,83765,83767],{"class":509,"line":683},[507,83766,514],{"class":513},[507,83768,80323],{"class":517},[507,83770,83771],{"class":509,"line":697},[507,83772,556],{"emptyLinePlaceholder":133},[507,83774,83775,83777,83779,83781,83783,83785,83787,83789,83792,83794,83796],{"class":509,"line":710},[507,83776,1370],{"class":513},[507,83778,73467],{"class":576},[507,83780,580],{"class":517},[507,83782,68762],{"class":1382},[507,83784,573],{"class":517},[507,83786,5682],{"class":583},[507,83788,622],{"class":517},[507,83790,83791],{"class":1382},"low_prob",[507,83793,573],{"class":517},[507,83795,72745],{"class":583},[507,83797,1883],{"class":517},[507,83799,83800,83802,83804,83806,83808,83810,83812,83814,83816],{"class":509,"line":715},[507,83801,73482],{"class":517},[507,83803,573],{"class":572},[507,83805,73487],{"class":517},[507,83807,22501],{"class":576},[507,83809,73492],{"class":517},[507,83811,68762],{"class":2155},[507,83813,573],{"class":572},[507,83815,73499],{"class":517},[507,83817,72708,83818],{"class":72706,"tabindex":72707},[507,83819,83820,83821,73508],{"class":72711,"role":72712},"Submits the circuit to real quantum hardware through Qollab. ",[504,83822,73507],{},[507,83824,83825],{"class":509,"line":721},[507,83826,556],{"emptyLinePlaceholder":133},[507,83828,83829],{"class":509,"line":736},[507,83830,81386],{"class":562},[507,83832,83833,83835,83837],{"class":509,"line":748},[507,83834,73513],{"class":513},[507,83836,64764],{"class":583},[507,83838,1728],{"class":517},[507,83840,83841,83843,83845,83847,83849],{"class":509,"line":761},[507,83842,81399],{"class":517},[507,83844,573],{"class":572},[507,83846,23993],{"class":517},[507,83848,73518],{"class":576},[507,83850,781],{"class":517},[507,83852,83853,83855,83857,83859,83861,83863,83865,83867,83869],{"class":509,"line":775},[507,83854,64185],{"class":572},[507,83856,580],{"class":517},[507,83858,22278],{"class":513},[507,83860,81418],{"class":730},[507,83862,2810],{"class":583},[507,83864,73518],{"class":517},[507,83866,2872],{"class":583},[507,83868,22281],{"class":730},[507,83870,587],{"class":517},[507,83872,83873,83875,83877,83879,83881,83883],{"class":509,"line":784},[507,83874,1734],{"class":513},[507,83876,81435],{"class":517},[507,83878,37008],{"class":513},[507,83880,73527],{"class":517},[507,83882,73530],{"class":583},[507,83884,1728],{"class":517},[507,83886,83887],{"class":509,"line":796},[507,83888,81448],{"class":513},[507,83890,83891,83893,83895,83897,83899],{"class":509,"line":809},[507,83892,73537],{"class":517},[507,83894,73540],{"class":576},[507,83896,580],{"class":517},[507,83898,23805],{"class":583},[507,83900,587],{"class":517},[507,83902,83903],{"class":509,"line":1352},[507,83904,556],{"emptyLinePlaceholder":133},[507,83906,83907,83909,83911,83913,83915],{"class":509,"line":1357},[507,83908,73551],{"class":517},[507,83910,573],{"class":572},[507,83912,23993],{"class":517},[507,83914,73558],{"class":576},[507,83916,781],{"class":517},[507,83918,83919],{"class":509,"line":1362},[507,83920,83921],{"class":562},"    # Filter low-probability noise\n",[507,83923,83924,83926,83928,83930,83932,83934,83936,83938,83940,83942,83944,83946,83948,83950,83952,83955],{"class":509,"line":1367},[507,83925,73551],{"class":517},[507,83927,573],{"class":572},[507,83929,81517],{"class":517},[507,83931,1630],{"class":513},[507,83933,81522],{"class":517},[507,83935,1636],{"class":513},[507,83937,73595],{"class":517},[507,83939,22607],{"class":576},[507,83941,1677],{"class":517},[507,83943,1645],{"class":513},[507,83945,81535],{"class":517},[507,83947,1651],{"class":572},[507,83949,69203],{"class":517},[507,83951,2391],{"class":572},[507,83953,83954],{"class":517}," low_prob}",[507,83956,72708,83957],{"class":72706,"tabindex":72707},[507,83958,83959],{"class":72711,"role":72712},"Drops outcomes that show up too rarely to be signal: a simple hardware-noise filter.",[507,83961,83962,83964,83966,83968,83970,83972,83974,83976,83978,83980,83982,83984,83986],{"class":509,"line":1379},[507,83963,2060],{"class":572},[507,83965,580],{"class":517},[507,83967,22278],{"class":513},[507,83969,81556],{"class":730},[507,83971,2810],{"class":583},[507,83973,68762],{"class":517},[507,83975,2872],{"class":583},[507,83977,81565],{"class":730},[507,83979,2810],{"class":583},[507,83981,81570],{"class":517},[507,83983,2872],{"class":583},[507,83985,22281],{"class":730},[507,83987,587],{"class":517},[74616,83989,83991],{"lead":83990},"QuantumCanvas is open source, built to be forked and extended.",[41852,83992,83993,84001],{},[41855,83994,83995],{},[41858,83996,83997,83999],{},[41861,83998,74627],{},[41861,84000,74630],{},[41868,84002,84003,84010,84018,84025],{},[41858,84004,84005,84007],{},[41873,84006,80076],{},[41873,84008,84009],{},"React + Canvas API, the interactive grid.",[41858,84011,84012,84015],{},[41873,84013,84014],{},"Backend",[41873,84016,84017],{},"Python \u002F FastAPI, turns the visual map into runnable circuits.",[41858,84019,84020,84022],{},[41873,84021,72215],{},[41873,84023,84024],{},"IonQ via Qollab, plus D-Wave Leap.",[41858,84026,84027,84030],{},[41873,84028,84029],{},"Infrastructure",[41873,84031,84032],{},"Microsoft Azure (AI Innovator Program).",[13,84034,84036],{"id":84035},"accessible-and-accurate","Accessible and accurate",[18,84038,84039],{},"The hardest part of the project was not the interface. It was calibration, deciding how much to simplify before the physics stops being the physics.",[72443,84041,84042],{"avatar":83522,"name":74240,"role":74335,"username":74336},[18,84043,84044],{},"Finding the balance between accessibility and accuracy. If you simplify too much, you lose what makes quantum mechanics meaningful. If you focus too heavily on technical rigor, the experience becomes inaccessible. Navigating that balance has been one of the most challenging and rewarding parts of building the project.",[18,84046,84047],{},"That tension is exactly why the primitives stay mapped to genuine quantum actions rather than becoming a purely classical abstraction. The goal is intuition you can carry back to real circuits, not a metaphor that falls apart the moment you leave the canvas.",[13,84049,73675],{"id":73674},[18,84051,84052],{},"QuantumCanvas is open for forking, and Shivani is most excited about people specializing it for real domains.",[72443,84054,84055],{"avatar":83522,"name":74240,"role":74335,"username":74336},[18,84056,84057],{},"I would love to see versions tailored to specific quantum applications. Areas like drug discovery, optimization, materials science, and logistics all present unique challenges and opportunities. It would be exciting to see people build specialized experiences that help others understand how quantum ideas connect to real-world problems.",[18,84059,84060],{},"The project's own roadmap maps that ambition onto three phases:",[42,84062,84063,84069,84075],{},[45,84064,84065,84068],{},[154,84066,84067],{},"Phase 1",": the core translation engine that turns a visual logic map into runnable hardware code.",[45,84070,84071,84074],{},[154,84072,84073],{},"Phase 2",": user testing with non-quantum engineers and students to refine the experience.",[45,84076,84077,84080],{},[154,84078,84079],{},"Phase 3",": a public beta and a template library for logistics, art, and chemistry.",[18,84082,84083,84084,10799,84088,53],{},"She is already extending the idea herself, with a second grid-based tool focused on quantum annealing and optimization that follows the same philosophy: make complex ideas easier to explore while keeping the intuition tied to the underlying science. For anyone just starting out, she points to two communities that moved her past reading and into building: the ",[49,84085,84087],{"href":84086},"https:\u002F\u002Fqiskit.org\u002Fevents\u002Fsummer-school\u002F","Qiskit Global Summer School",[49,84089,84091],{"href":84090},"https:\u002F\u002Fqworld.net","QWorld",[13,84093,73027],{"id":73026},[18,84095,84096],{},"If you have done the tutorials and want a place to actually build, QuantumCanvas is a good first canvas. It is open for forking on both Qollab and GitHub, with a live demo you can try right now.",[72443,84098,84099],{"avatar":83522,"name":74240,"role":74335,"username":74336},[18,84100,84101],{},"Looking back, QuantumCanvas has been as much a learning journey for me as it has been a tool for helping others learn quantum computing.",[73026,84103,84106],{"fork-href":83628,"live-href":84104,"title":84105},"https:\u002F\u002Fquantum-canvas-v1.netlify.app\u002F","Build a circuit by dragging, not typing.",[18,84107,84108,84109],{},"Fork QuantumCanvas, compose your own primitives, and run them on real hardware. ",[154,84110,73040],{},[953,84112,81675],{},{"title":104,"searchDepth":105,"depth":105,"links":84114},[84115,84116,84117,84118,84119],{"id":83527,"depth":105,"text":83528},{"id":83565,"depth":105,"text":83566},{"id":84035,"depth":105,"text":84036},{"id":73674,"depth":105,"text":73675},{"id":73026,"depth":105,"text":73027},[112,969,74228],[84122],{"username":74336,"name":74240,"role":84123,"avatar":83522,"bio":84124,"links":84125},"Quantum researcher & developer · Georgia Tech","Shivani is an M.S. computer-science researcher at Georgia Tech (Computing Systems), working across quantum computing, high-performance architecture, and human-computer interaction. She won the QRISE 2024 Infleqtion Challenge for work on VQE for atomic-clock precision, co-founded Qtangled, and has run quantum workshops for 250+ people. She has been building in quantum since 2020 and was selected for the D-Wave Leap Quantum LaunchPad in 2026.",[84126,84128,84129],{"label":73068,"href":84127},"https:\u002F\u002Fqollab.xyz\u002Fu\u002FShivaniMayekar",{"label":73059,"href":79368},{"label":979,"href":84130},"https:\u002F\u002Fgithub.com\u002FShivaniDM",{"username":8,"name":73075,"role":73076,"avatar":73077},"Shivani Mayekar built a visual sandbox where you drag and drop quantum operations to compose new algorithms, then run them on real hardware.","Shivani Mayekar built a visual sandbox to drag and drop quantum operations into algorithms and run them on real hardware. A Qollab Creative Challenge project.",{"href":83628,"label":84135},"Fork QuantumCanvas",{"image":83613,"alt":84137,"liveUrl":84104},"QuantumCanvas, a visual sandbox for composing quantum algorithms by hand",{},"\u002Fblog\u002Fquantum-canvas","2026-04-24",[],[84143,84144,84145],{"username":997,"project":998,"title":999,"category":73752,"thumb":1001,"to":1002},{"username":73742,"project":73743,"title":73744,"category":73097,"thumb":73745,"to":73746},{"username":74914,"project":76253,"title":74523,"category":75052,"thumb":76254,"to":74522},{"title":84147,"description":84148},"Quantum Creative Project Showcase: QuantumCanvas","A visual sandbox for building quantum algorithms by hand and running them on real hardware. Built by Shivani Mayekar for Qollab's Creative Challenge.","blog\u002Fquantum-canvas",[74477,143,84151],"tools","DqFZmEVOAlfEyN6FrYEb39lcrI2zPJDMtEPw7Q4Jn2M",{"id":84154,"title":84155,"authors":84156,"body":84157,"breadcrumb":84581,"builders":84583,"byline":116,"challenge":84584,"courseAuthor":116,"courseLead":116,"dek":84589,"description":84590,"draft":125,"extension":126,"eyebrow":84591,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":84592,"heroImage":116,"kind":131,"lessonCount":116,"meta":84596,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":84597,"publishDate":84598,"readingTime":116,"related":84599,"relatedProjects":116,"seo":84600,"stem":84603,"tags":84604,"track":116,"trackName":116,"__hash__":84605},"blog\u002Fblog\u002Fcreative-challenge.md","Quantum Creative Challenge",[8],{"type":10,"value":84158,"toc":84568},[84159,84162,84166,84219,84227,84231,84235,84238,84280,84288,84292,84295,84309,84312,84316,84331,84335,84338,84362,84365,84369,84372,84386,84390,84393,84530,84533,84537,84540,84547,84549,84553],[18,84160,84161],{},"Qollab is a community of developers, designers, and creative technologists figuring out what quantum software looks like by actually making things and sharing every step in the open. If you have got an idea, come build it with us.",[13,84163,84165],{"id":84164},"project-requirements","Project requirements",[84167,84168,84169,84193],"req-grid",{},[84170,84171,84173],"req-col",{"title":84172},"Technical",[42,84174,84175,84178,84181,84184,84187,84190],{},[45,84176,84177],{},"Use the Qiskit framework.",[45,84179,84180],{},"Open source, hosted in a public repository (for example, GitHub).",[45,84182,84183],{},"Listed on Qollab with runnable code examples.",[45,84185,84186],{},"Documentation sufficient for others to understand, replicate, or extend the project.",[45,84188,84189],{},"Basic accessibility standards (for example, color contrast and legible fonts for UI projects).",[45,84191,84192],{},"Publicly available for at least 12 months after completion.",[84170,84194,84196,84199],{"title":84195},"Creative",[18,84197,84198],{},"Projects should leverage quantum's unique attributes. We are looking for work that:",[42,84200,84201,84204,84207,84210,84213,84216],{},[45,84202,84203],{},"Demonstrates quantum speedups, especially across exponentially large solution spaces.",[45,84205,84206],{},"Uses superposition, entanglement, interference, or probabilistic outputs as part of its storytelling.",[45,84208,84209],{},"Inspires curiosity through design, interactivity, or narrative.",[45,84211,84212],{},"Shows creativity, originality, and community value.",[45,84214,84215],{},"Encourages others to learn from or remix the work.",[45,84217,84218],{},"Bonus points for quirky and playful.",[74178,84220,84221],{},[18,84222,84223,84226],{},[154,84224,84225],{},"Note:"," Building on existing projects is fine, provided your proposal includes a meaningful new twist and the original work is either yours, explicitly permitted, or open-source licensed.",[13,84228,84230],{"id":84229},"submission-and-timeline","Submission and timeline",[2513,84232,84234],{"id":84233},"what-to-submit","What to submit",[18,84236,84237],{},"Each proposal must include:",[42,84239,84240,84245,84251,84257,84263,84268,84274],{},[45,84241,84242],{},[154,84243,84244],{},"Project title.",[45,84246,84247,84250],{},[154,84248,84249],{},"Concept description:"," what you plan to build (up to 200 words).",[45,84252,84253,84256],{},[154,84254,84255],{},"Technical approach:"," how quantum computing will be used, plus other technologies involved.",[45,84258,84259,84262],{},[154,84260,84261],{},"Creative or educational value:"," how it contributes to the community or field (up to 200 words).",[45,84264,84265],{},[154,84266,84267],{},"Budget and compute resource estimates.",[45,84269,84270,84273],{},[154,84271,84272],{},"Timeline and deliverables"," through June 2026.",[45,84275,84276,84279],{},[154,84277,84278],{},"Team information:"," bios and highlights (up to 200 words per member).",[74178,84281,84282],{},[18,84283,84284,84287],{},[154,84285,84286],{},"Deadline:"," April 7th, 2026 via the application form. The call is now closed and is no longer accepting submissions.",[2513,84289,84291],{"id":84290},"review-criteria","Review criteria",[18,84293,84294],{},"Qollab and IonQ evaluate proposals on:",[42,84296,84297,84300,84303,84306],{},[45,84298,84299],{},"Creativity and originality.",[45,84301,84302],{},"Technical feasibility.",[45,84304,84305],{},"Clarity of idea and potential impact.",[45,84307,84308],{},"Alignment with our open, accessible ethos.",[18,84310,84311],{},"Spots are limited. We expect to fund only a handful of projects this round, so strong proposals that are clear, creative, and ready to execute will stand out. Selected participants were notified the week of April 13th, 2026.",[2513,84313,84315],{"id":84314},"delivery-timeline","Delivery timeline",[84317,84318],"timeline",{"l1":84319,"l2":84320,"l3":84321,"l4":84322,"l5":84323,"l6":84324,"w1":84325,"w2":84326,"w3":84327,"w4":84328,"w5":84329,"w6":84330},"Kickoff and initial payment","Midpoint check-in","Final delivery: code, docs, demo","QA review and final payment (50%)","Public launch and co-marketing","1-year maintenance ends","Apr 27","Wk of May 18","Jun 8","Wk of Jun 8","TBD","Jun 2027",[13,84332,84334],{"id":84333},"budget","Budget",[18,84336,84337],{},"Selected teams receive:",[42,84339,84340,84347,84353,84356,84359],{},[45,84341,84342,84343,84346],{},"Up to ",[154,84344,84345],{},"$5,000 USD"," (less applicable tax).",[45,84348,84342,84349,84352],{},[154,84350,84351],{},"$50,000 in IonQ compute credits",", solely for the funded project. Credits expire after 60 days. The first three funded projects receive an additional $20,000 in credits from IonQ, so early, strong proposals have a real advantage.",[45,84354,84355],{},"Promotion and a feature on our website.",[45,84357,84358],{},"Weekly support from IonQ experts.",[45,84360,84361],{},"Virtual access to IonQ executives to present your work.",[18,84363,84364],{},"Participants must provide a payment method and are solely responsible for tax reporting.",[13,84366,84368],{"id":84367},"additional-requirements","Additional requirements",[18,84370,84371],{},"Selected participants agree to:",[42,84373,84374,84377,84380,84383],{},[45,84375,84376],{},"Post at least once on their social channels when their project goes live on Qollab.xyz.",[45,84378,84379],{},"Allow Qollab and IonQ to feature their work and profile.",[45,84381,84382],{},"Release code and documentation under an MIT license. IP remains with the creator.",[45,84384,84385],{},"Include the attribution: \"This effort is supported via compute credits from Qollab and IonQ.\"",[13,84387,84389],{"id":84388},"eligibility","Eligibility",[18,84391,84392],{},"Open to participants based in:",[84394,84395,84396],"countries",{},[42,84397,84398,84401,84404,84407,84410,84413,84416,84419,84422,84425,84428,84431,84434,84437,84440,84443,84446,84449,84452,84455,84458,84461,84464,84467,84470,84473,84476,84479,84482,84485,84488,84491,84494,84497,84500,84503,84506,84509,84512,84515,84518,84521,84524,84527],{},[45,84399,84400],{},"Argentina",[45,84402,84403],{},"Australia",[45,84405,84406],{},"Austria",[45,84408,84409],{},"Belgium",[45,84411,84412],{},"Brazil",[45,84414,84415],{},"Bulgaria",[45,84417,84418],{},"Canada",[45,84420,84421],{},"Chile",[45,84423,84424],{},"Colombia",[45,84426,84427],{},"Cyprus",[45,84429,84430],{},"Czechia",[45,84432,84433],{},"Denmark",[45,84435,84436],{},"Estonia",[45,84438,84439],{},"Finland",[45,84441,84442],{},"France",[45,84444,84445],{},"Germany",[45,84447,84448],{},"Greece",[45,84450,84451],{},"Hungary",[45,84453,84454],{},"India",[45,84456,84457],{},"Indonesia",[45,84459,84460],{},"Ireland",[45,84462,84463],{},"Israel",[45,84465,84466],{},"Italy",[45,84468,84469],{},"Japan",[45,84471,84472],{},"Latvia",[45,84474,84475],{},"Lithuania",[45,84477,84478],{},"Luxembourg",[45,84480,84481],{},"Malaysia",[45,84483,84484],{},"Malta",[45,84486,84487],{},"Mexico",[45,84489,84490],{},"Netherlands",[45,84492,84493],{},"New Zealand",[45,84495,84496],{},"Norway",[45,84498,84499],{},"Poland",[45,84501,84502],{},"Portugal",[45,84504,84505],{},"Romania",[45,84507,84508],{},"Singapore",[45,84510,84511],{},"Slovakia",[45,84513,84514],{},"South Korea",[45,84516,84517],{},"Spain",[45,84519,84520],{},"Sweden",[45,84522,84523],{},"Switzerland",[45,84525,84526],{},"United Kingdom",[45,84528,84529],{},"United States",[18,84531,84532],{},"Individuals and teams may submit multiple proposals. Credits and funding are awarded per project, not per person.",[13,84534,84536],{"id":84535},"examples","Examples",[18,84538,84539],{},"We previously ran a challenge in Winter of 2025. These are the two projects that came out of it:",[84541,84542],"example-grid",{"d1":84543,"d2":84544,"i1":84545,"i2":1001,"t1":1006,"t2":84546,"u1":1009,"u2":1002},"An interactive generative art installation by Amber Pincar and Justin Pincar, where digital plants exist in quantum superposition until observed.","A scientifically rigorous quantum circuit and algorithm visualization engine built for teaching and conceptual clarity.","\u002F_content\u002Fimages\u002Fquantum-garden\u002Ffig1-garden-preview.webp","Quantum Algorithm Visualization Engine (QAVE)",[13,84548,85],{"id":84},[18,84550,88,84551,53],{},[49,84552,92],{"href":91},[94,84554,84558],{"f1":74443,"f2":79459,"l1":84555,"l2":84556,"title":84557},"See what people built","Back to news","This challenge is closed",[18,84559,84560,84561,84563,84564,84567],{},"The submission window has ended and we are no longer accepting proposals. If you have a project you would like to talk to us about, email ",[49,84562,92],{"href":91}," — and keep an eye on ",[49,84565,113],{"href":84566},"\u002Fprograms"," for the next open call.",{"title":104,"searchDepth":105,"depth":105,"links":84569},[84570,84571,84576,84577,84578,84579,84580],{"id":84164,"depth":105,"text":84165},{"id":84229,"depth":105,"text":84230,"children":84572},[84573,84574,84575],{"id":84233,"depth":540,"text":84234},{"id":84290,"depth":540,"text":84291},{"id":84314,"depth":540,"text":84315},{"id":84333,"depth":105,"text":84334},{"id":84367,"depth":105,"text":84368},{"id":84388,"depth":105,"text":84389},{"id":84535,"depth":105,"text":84536},{"id":84,"depth":105,"text":85},[112,969,84582],"Creative Challenge",[],{"type":84585,"status":84586,"deadline":73796,"prize":84587,"terms":84588},"open-call","closed","Up to $5,000 + $50,000 in IonQ compute credits","Open source (MIT), public for 12+ months","Quantum computing is buildable today. Not in five years, not in a research lab, but right now, in your browser. For the Creative Challenge, we backed a small number of projects with a monetary grant, IonQ compute credits, hands-on support, and a spotlight for their work.","The Qollab Creative Challenge funded developers to build on real IonQ hardware with Qiskit. The call is closed and no longer accepting proposals.","Submissions closed",{"primaryHref":74443,"primaryLabel":84593,"secondaryHref":84566,"secondaryLabel":84594,"note":84595},"See what people built →","All programs","The submission window closed on April 7th, 2026 and this call is no longer accepting proposals. The requirements below stay published for reference.",{},"\u002Fblog\u002Fcreative-challenge","2026-04-07",[],{"title":84601,"description":84602},"Quantum Creative Challenge: Build on IonQ | Qollab","Grants, IonQ compute credits, and a spotlight for open-source quantum projects. The call closed on April 7th, 2026 and is no longer accepting proposals.","blog\u002Fcreative-challenge",[84585,142,143],"6ZKTMzoyPXGpFXRtpW2b7KMxZAC9p2JVkEidKomAg9g",{"id":84607,"title":84608,"authors":84609,"body":84610,"breadcrumb":85169,"builders":85170,"byline":85179,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":85180,"description":85181,"draft":125,"extension":126,"eyebrow":85182,"finish":116,"fork":85183,"hero":85185,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":85187,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":85188,"publishDate":85189,"readingTime":993,"related":85190,"relatedProjects":85191,"seo":85196,"stem":85197,"tags":85198,"track":116,"trackName":116,"__hash__":85200},"blog\u002Fblog\u002Fqave.md","Project Showcase: QAVE",[997],{"type":10,"value":84611,"toc":85162},[84612,84615,84618,84621,84625,84628,84631,84636,84639,84643,84646,84649,84652,84657,84660,85092,85096,85099,85104,85107,85112,85115,85120,85123,85127,85130,85135,85138,85141,85146,85148,85151,85159],[18,84613,84614],{},"What does a quantum algorithm actually look like while it's running? Not the math on paper. The actual state, evolving gate by gate.",[18,84616,84617],{},"QAVE (Quantum Algorithm Visualization Engine) answers that question visually. QAVE computes the quantum state at every gate operation, synchronizes multiple views into a single coherent replay, and makes quantum errors and their corrections directly observable. Not a diagram. Not a static plot. A full animation of what's happening inside the circuit as it runs.",[18,84619,84620],{},"QAVE is a featured project from Qollab's first Quantum Creative Challenge. It addresses a gap that Inho noticed even with years of research experience: the distance between how algorithms are formally described and how intuitively they can be understood.",[13,84622,84624],{"id":84623},"making-the-invisible-visible","Making the invisible visible",[18,84626,84627],{},"Inho Choi is not new to quantum. He's a researcher working at the intersection of quantum physics and computation, drawn to fundamental open problems where new mathematical and algorithmic tools are still needed.",[18,84629,84630],{},"But deep expertise didn't shield him from a frustration shared by students and researchers alike: quantum algorithms are hard to follow, and not always because of the math.",[72443,84632,84634],{"avatar":84633,"name":74223,"role":74318,"username":997},"\u002F_content\u002Fimages\u002Fbuilders\u002Finho-choi.webp",[18,84635,74321],{},[18,84637,84638],{},"That observation became the starting point for QAVE. He wanted to close the gap between formal description and genuine understanding. Not by simplifying the science, but by giving it a visual layer that carries real information.",[13,84640,84642],{"id":84641},"how-inho-built-qave","How Inho built QAVE",[18,84644,84645],{},"The core design decision behind QAVE is that everything stays in sync. The engine computes the full quantum state at every step of a gate operation and locks multiple views together into one replay. State vector, density matrix, gate effects. All moving in lockstep.",[18,84647,84648],{},"That matters most when things go wrong. Quantum errors are physical events with specific signatures, and understanding how they propagate through a circuit is central to building reliable quantum systems. Most tools leave that as something you reconstruct from equations after the fact. QAVE makes it something you watch happen.",[18,84650,84651],{},"The result is a tool built for educators and researchers who want a transparent, grounded way to see how quantum algorithms actually behave. Not an approximation. The real evolution, made visible.",[831,84653],{"caption":84654,"no":835,"poster":84655,"video":84656},"A QAVE replay of a 5-qubit GHZ state: state vector, density matrix, and gate effects evolving in lockstep as the circuit runs.","\u002F_content\u002Fimages\u002Fqave\u002Fdemo-poster.webp","https:\u002F\u002Fapi.cms.qollab.xyz\u002Fassets\u002F50e7233c-a05b-4df8-802e-aa8b97fcc5c2",[18,84658,84659],{},"The whole workflow is a short, runnable notebook. This is the core of it: build a circuit, generate a deterministic trace, and render the synchronized animation you just watched.",[493,84661,84664],{"name":84662,"run-href":84663,"tag":496},"ghz3_with_QAVE.ipynb","\u002Fu\u002Fq-inho\u002Fqave",[498,84665,84667],{"className":500,"code":84666,"language":502,"meta":104,"style":104},"# QAVE turns a Qiskit circuit into a deterministic, replayable animation.\nfrom qiskit import QuantumCircuit\nfrom qiskit.quantum_info import Statevector\nfrom qave import SimulationOptions, generate_trace_from_qiskit\nfrom qave.notebook import render_animation, display_animation, resolve_notebook_render_options\nfrom IPython.display import display\n\n# Build a 3-qubit GHZ circuit\ncircuit = QuantumCircuit(3, 3, name=\"ghz3\")\ncircuit.h(0)\ncircuit.cx(0, 1)\ncircuit.cx(0, 2)\ncircuit.measure(range(3), range(3))\n\n# Confirm an equal superposition of |000⟩ and |111⟩ before measuring\npsi = Statevector.from_instruction(circuit.remove_final_measurements(inplace=False))\n\n# Generate a deterministic trace: the state computed at every step of every gate\noptions = SimulationOptions(algorithm_id=\"ghz\", mode=\"preview\", seed=24, shot_count=100)\nresult = generate_trace_from_qiskit(circuit, options=options)\n\n# Deterministic measurement replay: same seed, same shots, every time\nreplay = result.require_measurement_shot_replay()\n\n# Render the synchronized circuit \u002F amplitude \u002F probability animation\nrender = resolve_notebook_render_options(width=1280, height=720, fps=60)\nanim = render_animation(result, render=render)\ndisplay(display_animation(anim))\n",[504,84668,84669,84674,84684,84695,84707,84719,84730,84734,84739,84766,84785,84809,84830,84854,84858,84863,84892,84896,84901,84949,84976,84980,84985,85005,85009,85014,85055,85075],{"__ignoreMap":104},[507,84670,84671],{"class":509,"line":510},[507,84672,84673],{"class":562},"# QAVE turns a Qiskit circuit into a deterministic, replayable animation.\n",[507,84675,84676,84678,84680,84682],{"class":509,"line":105},[507,84677,529],{"class":513},[507,84679,532],{"class":517},[507,84681,514],{"class":513},[507,84683,537],{"class":517},[507,84685,84686,84688,84690,84692],{"class":509,"line":540},[507,84687,529],{"class":513},[507,84689,545],{"class":517},[507,84691,514],{"class":513},[507,84693,84694],{"class":517}," Statevector\n",[507,84696,84697,84699,84702,84704],{"class":509,"line":553},[507,84698,529],{"class":513},[507,84700,84701],{"class":517}," qave ",[507,84703,514],{"class":513},[507,84705,84706],{"class":517}," SimulationOptions, generate_trace_from_qiskit\n",[507,84708,84709,84711,84714,84716],{"class":509,"line":559},[507,84710,529],{"class":513},[507,84712,84713],{"class":517}," qave.notebook ",[507,84715,514],{"class":513},[507,84717,84718],{"class":517}," render_animation, display_animation, resolve_notebook_render_options\n",[507,84720,84721,84723,84725,84727],{"class":509,"line":566},[507,84722,529],{"class":513},[507,84724,67822],{"class":517},[507,84726,514],{"class":513},[507,84728,84729],{"class":517}," display\n",[507,84731,84732],{"class":509,"line":590},[507,84733,556],{"emptyLinePlaceholder":133},[507,84735,84736],{"class":509,"line":610},[507,84737,84738],{"class":562},"# Build a 3-qubit GHZ circuit\n",[507,84740,84741,84743,84745,84747,84749,84751,84753,84755,84757,84759,84761,84764],{"class":509,"line":634},[507,84742,73339],{"class":517},[507,84744,573],{"class":572},[507,84746,577],{"class":576},[507,84748,580],{"class":517},[507,84750,8226],{"class":583},[507,84752,622],{"class":517},[507,84754,8226],{"class":583},[507,84756,622],{"class":517},[507,84758,22008],{"class":2155},[507,84760,573],{"class":572},[507,84762,84763],{"class":730},"\"ghz3\"",[507,84765,587],{"class":517},[507,84767,84768,84770,84772,84774,84776,84778],{"class":509,"line":661},[507,84769,73358],{"class":517},[507,84771,596],{"class":576},[507,84773,580],{"class":517},[507,84775,601],{"class":583},[507,84777,3649],{"class":517},[507,84779,72708,84780],{"class":72706,"tabindex":72707},[507,84781,84782,84784],{"class":72711,"role":72712},[154,84783,73373],{}," Puts qubit 0 into an equal mix of |0⟩ and |1⟩.",[507,84786,84787,84789,84791,84793,84795,84797,84799,84801],{"class":509,"line":678},[507,84788,73358],{"class":517},[507,84790,615],{"class":576},[507,84792,580],{"class":517},[507,84794,601],{"class":583},[507,84796,622],{"class":517},[507,84798,625],{"class":583},[507,84800,3649],{"class":517},[507,84802,72708,84803],{"class":72706,"tabindex":72707},[507,84804,84805,84808],{"class":72711,"role":72712},[154,84806,84807],{},"Entangle."," Links qubit 1 to qubit 0.",[507,84810,84811,84813,84815,84817,84819,84821,84823,84825],{"class":509,"line":683},[507,84812,73358],{"class":517},[507,84814,615],{"class":576},[507,84816,580],{"class":517},[507,84818,601],{"class":583},[507,84820,622],{"class":517},[507,84822,584],{"class":583},[507,84824,3649],{"class":517},[507,84826,72708,84827],{"class":72706,"tabindex":72707},[507,84828,84829],{"class":72711,"role":72712},"Entangles qubit 2 as well: the full GHZ state |000⟩ + |111⟩.",[507,84831,84832,84834,84836,84838,84840,84842,84844,84846,84848,84850,84852],{"class":509,"line":697},[507,84833,73358],{"class":517},[507,84835,72822],{"class":576},[507,84837,580],{"class":517},[507,84839,2204],{"class":572},[507,84841,580],{"class":517},[507,84843,8226],{"class":583},[507,84845,2213],{"class":517},[507,84847,2204],{"class":572},[507,84849,580],{"class":517},[507,84851,8226],{"class":583},[507,84853,22540],{"class":517},[507,84855,84856],{"class":509,"line":710},[507,84857,556],{"emptyLinePlaceholder":133},[507,84859,84860],{"class":509,"line":715},[507,84861,84862],{"class":562},"# Confirm an equal superposition of |000⟩ and |111⟩ before measuring\n",[507,84864,84865,84868,84870,84873,84876,84879,84882,84884,84886,84888,84890],{"class":509,"line":721},[507,84866,84867],{"class":517},"psi ",[507,84869,573],{"class":572},[507,84871,84872],{"class":517}," Statevector.",[507,84874,84875],{"class":576},"from_instruction",[507,84877,84878],{"class":517},"(circuit.",[507,84880,84881],{"class":576},"remove_final_measurements",[507,84883,580],{"class":517},[507,84885,13873],{"class":2155},[507,84887,573],{"class":572},[507,84889,21964],{"class":583},[507,84891,22540],{"class":517},[507,84893,84894],{"class":509,"line":736},[507,84895,556],{"emptyLinePlaceholder":133},[507,84897,84898],{"class":509,"line":748},[507,84899,84900],{"class":562},"# Generate a deterministic trace: the state computed at every step of every gate\n",[507,84902,84903,84906,84908,84911,84913,84916,84918,84921,84923,84925,84927,84930,84932,84934,84936,84938,84940,84943,84945,84947],{"class":509,"line":761},[507,84904,84905],{"class":517},"options ",[507,84907,573],{"class":572},[507,84909,84910],{"class":576}," SimulationOptions",[507,84912,580],{"class":517},[507,84914,84915],{"class":2155},"algorithm_id",[507,84917,573],{"class":572},[507,84919,84920],{"class":730},"\"ghz\"",[507,84922,622],{"class":517},[507,84924,23938],{"class":2155},[507,84926,573],{"class":572},[507,84928,84929],{"class":730},"\"preview\"",[507,84931,622],{"class":517},[507,84933,81883],{"class":2155},[507,84935,573],{"class":572},[507,84937,23683],{"class":583},[507,84939,622],{"class":517},[507,84941,84942],{"class":2155},"shot_count",[507,84944,573],{"class":572},[507,84946,5682],{"class":583},[507,84948,587],{"class":517},[507,84950,84951,84953,84955,84958,84960,84962,84964,84967],{"class":509,"line":775},[507,84952,78915],{"class":517},[507,84954,573],{"class":572},[507,84956,84957],{"class":576}," generate_trace_from_qiskit",[507,84959,73492],{"class":517},[507,84961,23946],{"class":2155},[507,84963,573],{"class":572},[507,84965,84966],{"class":517},"options)",[507,84968,72708,84969],{"class":72706,"tabindex":72707},[507,84970,84971,84972,84975],{"class":72711,"role":72712},"Emits a versioned ",[504,84973,84974],{},"trace.json"," with physically-computed in-gate evolution samples.",[507,84977,84978],{"class":509,"line":784},[507,84979,556],{"emptyLinePlaceholder":133},[507,84981,84982],{"class":509,"line":796},[507,84983,84984],{"class":562},"# Deterministic measurement replay: same seed, same shots, every time\n",[507,84986,84987,84990,84992,84995,84998,85000],{"class":509,"line":809},[507,84988,84989],{"class":517},"replay ",[507,84991,573],{"class":572},[507,84993,84994],{"class":517}," result.",[507,84996,84997],{"class":576},"require_measurement_shot_replay",[507,84999,66172],{"class":517},[507,85001,72708,85002],{"class":72706,"tabindex":72707},[507,85003,85004],{"class":72711,"role":72712},"Repeatable shot outcomes, so the animation replays identically every run.",[507,85006,85007],{"class":509,"line":1352},[507,85008,556],{"emptyLinePlaceholder":133},[507,85010,85011],{"class":509,"line":1357},[507,85012,85013],{"class":562},"# Render the synchronized circuit \u002F amplitude \u002F probability animation\n",[507,85015,85016,85019,85021,85024,85026,85029,85031,85034,85036,85039,85041,85044,85046,85049,85051,85053],{"class":509,"line":1362},[507,85017,85018],{"class":517},"render ",[507,85020,573],{"class":572},[507,85022,85023],{"class":576}," resolve_notebook_render_options",[507,85025,580],{"class":517},[507,85027,85028],{"class":2155},"width",[507,85030,573],{"class":572},[507,85032,85033],{"class":583},"1280",[507,85035,622],{"class":517},[507,85037,85038],{"class":2155},"height",[507,85040,573],{"class":572},[507,85042,85043],{"class":583},"720",[507,85045,622],{"class":517},[507,85047,85048],{"class":2155},"fps",[507,85050,573],{"class":572},[507,85052,58677],{"class":583},[507,85054,587],{"class":517},[507,85056,85057,85060,85062,85065,85068,85070,85072],{"class":509,"line":1367},[507,85058,85059],{"class":517},"anim ",[507,85061,573],{"class":572},[507,85063,85064],{"class":576}," render_animation",[507,85066,85067],{"class":517},"(result, ",[507,85069,60312],{"class":2155},[507,85071,573],{"class":572},[507,85073,85074],{"class":517},"render)\n",[507,85076,85077,85079,85081,85084,85087],{"class":509,"line":1379},[507,85078,70479],{"class":576},[507,85080,580],{"class":517},[507,85082,85083],{"class":576},"display_animation",[507,85085,85086],{"class":517},"(anim))",[507,85088,72708,85089],{"class":72706,"tabindex":72707},[507,85090,85091],{"class":72711,"role":72712},"The synchronized replay above (Fig. 1), rendered to MP4 with a GIF fallback.",[13,85093,85095],{"id":85094},"where-others-could-take-it","Where others could take it",[18,85097,85098],{},"QAVE is open for forking. When asked what directions excite him most, Inho pointed to several.",[72443,85100,85101],{"avatar":84633,"name":74223,"role":74318,"username":997},[18,85102,85103],{},"The near-term add-ons that excite me most are already visible in the roadmap: a realistic noise mode so users can compare ideal and hardware-like behavior, a tensor-network or MPS view that makes larger circuits and entanglement structure easier to explore, and a hybrid-loop mode for VQE or QML that shows how the quantum circuit and the classical optimizer co-evolve over many iterations.",[18,85105,85106],{},"Longer term, Inho is excited by a hardware-aware layer on top of that.",[72443,85108,85109],{"avatar":84633,"name":74223,"role":74318,"username":997},[18,85110,85111],{},"Visualizing how an algorithm is implemented on different platforms and how noise propagates from the physical device to the final algorithmic performance.",[18,85113,85114],{},"He also has a clear picture of what a companion project could look like.",[72443,85116,85117],{"avatar":84633,"name":74223,"role":74318,"username":997},[18,85118,85119],{},"If I built a companion project on Qollab with the same setup, it would probably focus on the hardware layer itself. A visual explorer or lightweight simulator that compares how the same logical circuit is realized on superconducting, trapped-ion, and neutral-atom platforms, and how they affect performance end to end. I think there is real value in helping people see how quantum computing is implemented and run on different physical hardware.",[18,85121,85122],{},"The foundation is there. Someone with the right interest could pick up any of these threads.",[13,85124,85126],{"id":85125},"finding-your-way-in","Finding your way in",[18,85128,85129],{},"Inho's connection to quantum runs deeper than a research interest. It started with a question about computation and nature.",[72443,85131,85132],{"avatar":84633,"name":74223,"role":74318,"username":997},[18,85133,85134],{},"What made me feel that I belonged in quantum was that it spoke directly to a question I had already cared about for a long time: how to understand and simulate nature faithfully. Quantum computation felt like the framework that resolves the bottleneck. As Feynman put it, if nature is fundamentally quantum, then simulating it properly may require computation that is quantum as well.",[18,85136,85137],{},"That idea stayed with him from the beginning. It made quantum computing feel like a natural path, not a career pivot. And it shapes how he thinks about accessibility. He believes conceptual clarity and scientific rigor should go together, and that how we teach quantum science matters as much as the science itself. QAVE is that belief turned into a tool.",[18,85139,85140],{},"Inho's advice for finding community in quantum is simple:",[72443,85142,85143],{"avatar":84633,"name":74223,"role":74318,"username":997},[18,85144,85145],{},"Start small and be visible. Find an active recurring group, whether that is a local seminar, an online study group, or an open-source project, and participate consistently. You do not need to impress people. You just need to engage honestly with the material. In my experience, community starts to form when you stop trying to learn everything alone and start sharing your questions, your attempts, and your progress with others.",[13,85147,73027],{"id":73026},[18,85149,85150],{},"If you're curious about what quantum circuits actually do at each step, QAVE is a good place to start. The project code is open for forking on both Qollab and GitHub.",[73026,85152,85154],{"fork-href":84663,"title":85153},"See quantum algorithms run, gate by gate.",[18,85155,85156,85157],{},"Fork QAVE and explore the state evolution yourself, or pick up one of the roadmap threads above. ",[154,85158,73040],{},[953,85160,85161],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":85163},[85164,85165,85166,85167,85168],{"id":84623,"depth":105,"text":84624},{"id":84641,"depth":105,"text":84642},{"id":85094,"depth":105,"text":85095},{"id":85125,"depth":105,"text":85126},{"id":73026,"depth":105,"text":73027},[112,969,999],[85171],{"username":997,"name":74223,"role":85172,"avatar":84633,"bio":85173,"links":85174},"Quantum information researcher","A researcher in quantum information theory and many-body physics, drawn to the open problems where new mathematical and algorithmic tools are still needed. Inho bridges theoretical foundations and practical implementation, and believes conceptual clarity and scientific rigor belong together.",[85175,85177],{"label":73068,"href":85176},"\u002Fu\u002Fq-inho",{"label":73059,"href":85178},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Finho-choi01\u002F",{"username":8,"name":73075,"role":73076,"avatar":73077},"Inho Choi built a visualization engine that makes quantum algorithms visible, step by step.","What does a quantum algorithm look like while it runs? QAVE makes the state evolve gate by gate. A featured project from Qollab's Creative Challenge.","Quantum Creative Challenge · Fall 2025",{"href":84663,"label":85184},"Fork QAVE",{"image":1001,"alt":85186},"QAVE, Quantum Algorithm Visualization Engine",{},"\u002Fblog\u002Fqave","2026-04-03",[],[85192,85193,85195],{"username":74336,"project":77810,"title":74228,"category":73752,"thumb":77811,"to":74329},{"username":74354,"project":85194,"title":74245,"category":73752,"thumb":80168,"to":74347},"quantum-game-pizza-race",{"username":1004,"project":1005,"title":1006,"category":1007,"thumb":1008,"to":1009},{"title":84608,"description":85181},"blog\u002Fqave",[1019,143,85199],"featured","ioFbHu2ihQhfzSeOsaAsFB0Z14q6oVOkttT7c0i3H6c",{"id":85202,"title":85203,"authors":85204,"body":85205,"breadcrumb":85720,"builders":85721,"byline":85736,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":85737,"description":85738,"draft":125,"extension":126,"eyebrow":85182,"finish":116,"fork":85739,"hero":85740,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":73085,"lessonCount":116,"meta":85742,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":116,"path":85743,"publishDate":85744,"readingTime":80935,"related":85745,"relatedProjects":85746,"seo":85750,"stem":85753,"tags":85754,"track":116,"trackName":116,"__hash__":85755},"blog\u002Fblog\u002Fquantum-garden.md","Project Showcase: Quantum Garden",[1004],{"type":10,"value":85206,"toc":85713},[85207,85210,85214,85217,85220,85224,85227,85230,85234,85238,85241,85244,85247,85250,85253,85257,85259,85262,85265,85600,85604,85615,85619,85624,85671,85675,85678,85684,85690,85696,85698,85701,85710],[18,85208,85209],{},"Quantum Garden is a generative art exhibit where digital plants get their form, color, and behavior from circuits run on real IonQ quantum hardware. Not simulated randomness, but actual quantum measurement outcomes baked permanently into each plant. Built by data scientist Amber Wang and engineer Justin Pincar for Qollab's Quantum Creative Challenge, it reframes quantum computing as a creative medium rather than a research tool, and makes the case that meaningful quantum work no longer requires a physics PhD. Just a strong concept and access to cloud hardware.",[13,85211,85213],{"id":85212},"from-out-of-reach-to-real-quantum","From \"out of reach\" to real quantum",[18,85215,85216],{},"Neither Amber Wang nor Justin Pincar are quantum researchers. That's part of what makes this story even more interesting.",[18,85218,85219],{},"Amber is a data scientist, SEO strategist, and co-founder of PressRoom AI. She has been thinking about probability, uncertainty, and complex systems across her work in commercial real-estate analytics and AI. She's also a garden lover, which is where the concept started.",[72443,85221,85222],{"avatar":73979,"name":73980,"role":73981,"username":1004},[18,85223,73984],{},[18,85225,85226],{},"Justin is a software engineer and CTO of Achievable, with a background that includes open-sourcing AdWhirl at Google and scaling it to over a billion ad impressions per day. He had been tracking quantum from a distance for years. Interested, but treating it as out of reach.",[18,85228,85229],{},"What changed that? Access. Within a few months they were able to build out their quantum art exhibit and present it to the world.",[72443,85231,85232],{"avatar":73990,"name":73991,"role":73981,"username":1004},[18,85233,73994],{},[13,85235,85237],{"id":85236},"the-idea-behind-quantum-garden","The idea behind Quantum Garden",[18,85239,85240],{},"It started with a feeling. Amber, who co-created the project, has always found deep joy and peace in real gardens; the slow pace of them, the way things grow and fade without asking permission. Justin brought the technical imagination.",[18,85242,85243],{},"Together they asked: what if a garden could be powered by the actual randomness of quantum physics? Not simulated randomness. Real quantum measurement outcomes, run on real hardware, woven permanently into every plant.",[18,85245,85246],{},"The result is an experience that teaches through presence rather than instruction. You're discovering something that was already true instead of making a choice whenever you hover over a plant.",[18,85248,85249],{},"Some plants are entangled with others across the garden; observing one reveals something about a plant you haven't visited yet. A quiet panel slides in afterward, showing the quantum circuit that shaped what you just saw, never intrusive, always dismissible.",[18,85251,85252],{},"The garden doesn't wait for you. It germinates, blooms, and fades on its own timeline. You can step away for a week and come back to find it has gone through something like a season.",[831,85254],{"alt":85255,"caption":85256,"no":835,"src":84545},"Garden preview showing 131 digital plants scattered across a soft lavender canvas, with a detail panel open for a selected plant called 'Luminous Tulip'","An interactive garden view: 131 digital plants scattered across a soft lavender-white canvas, with a detail panel open for a selected plant called \"Luminous Tulip\".",[13,85258,77017],{"id":77016},[18,85260,85261],{},"Quantum Garden is built around a core design decision: instead of running quantum circuits live when a visitor arrives (which would be slow and expensive), the team pre-computes a pool of quantum measurement results in advance. Each plant draws from that pool when it's first observed.",[18,85263,85264],{},"This turned a hardware constraint, quantum circuit execution latency, into a feature. The delay became germination. The asynchronous nature of real quantum runs became the reason the garden has its own sense of time.",[493,85266,85269],{"name":85267,"run-href":85268,"tag":496},"plant_circuit.py","\u002Fu\u002FAmberPincar\u002Fquantum-garden",[498,85270,85272],{"className":500,"code":85271,"language":502,"meta":104,"style":104},"from qiskit import QuantumCircuit\nimport math\n\n# Each plant gets a deterministic seed from its ID hash\nseed = 42\nqc = QuantumCircuit(5, 5)\n\n# 1 · Full superposition — all 32 outcomes are possible\nfor i in range(5):\n    qc.h(i)\n\n# 2 · Seed-based Ry rotations — a unique bias per qubit\nfor i in range(5):\n    qc.ry((seed * (i + 1) * 0.1) % (2 * math.pi), i)\n\n# 3 · Entanglement chain — correlate neighbouring qubits\nfor i in range(4):\n    qc.cx(i, i + 1)\n\n# 4 · Cross-entanglement — non-local correlations\nqc.cx(0, 2); qc.cx(1, 3); qc.cx(2, 4)\n\n# Measure → the plant's permanent quantum fingerprint\nqc.measure(range(5), range(5))\ncounts = backend.run(qc, shots=100).result().get_counts()\n",[504,85273,85274,85284,85290,85294,85299,85309,85327,85331,85336,85352,85367,85371,85376,85392,85435,85439,85444,85460,85482,85486,85491,85532,85536,85541,85565],{"__ignoreMap":104},[507,85275,85276,85278,85280,85282],{"class":509,"line":510},[507,85277,529],{"class":513},[507,85279,532],{"class":517},[507,85281,514],{"class":513},[507,85283,537],{"class":517},[507,85285,85286,85288],{"class":509,"line":105},[507,85287,514],{"class":513},[507,85289,57135],{"class":517},[507,85291,85292],{"class":509,"line":540},[507,85293,556],{"emptyLinePlaceholder":133},[507,85295,85296],{"class":509,"line":553},[507,85297,85298],{"class":562},"# Each plant gets a deterministic seed from its ID hash\n",[507,85300,85301,85304,85306],{"class":509,"line":559},[507,85302,85303],{"class":517},"seed ",[507,85305,573],{"class":572},[507,85307,85308],{"class":583}," 42\n",[507,85310,85311,85313,85315,85317,85319,85321,85323,85325],{"class":509,"line":566},[507,85312,569],{"class":517},[507,85314,573],{"class":572},[507,85316,577],{"class":576},[507,85318,580],{"class":517},[507,85320,58245],{"class":583},[507,85322,622],{"class":517},[507,85324,58245],{"class":583},[507,85326,587],{"class":517},[507,85328,85329],{"class":509,"line":590},[507,85330,556],{"emptyLinePlaceholder":133},[507,85332,85333],{"class":509,"line":610},[507,85334,85335],{"class":562},"# 1 · Full superposition — all 32 outcomes are possible\n",[507,85337,85338,85340,85342,85344,85346,85348,85350],{"class":509,"line":634},[507,85339,1630],{"class":513},[507,85341,8246],{"class":517},[507,85343,1636],{"class":513},[507,85345,8221],{"class":572},[507,85347,580],{"class":517},[507,85349,58245],{"class":583},[507,85351,1883],{"class":517},[507,85353,85354,85356,85358,85360],{"class":509,"line":661},[507,85355,21867],{"class":517},[507,85357,596],{"class":576},[507,85359,83031],{"class":517},[507,85361,72708,85362],{"class":72706,"tabindex":72707},[507,85363,85364,85366],{"class":72711,"role":72712},[154,85365,73373],{}," A Hadamard on every qubit puts all 32 outcomes into play at once.",[507,85368,85369],{"class":509,"line":678},[507,85370,556],{"emptyLinePlaceholder":133},[507,85372,85373],{"class":509,"line":683},[507,85374,85375],{"class":562},"# 2 · Seed-based Ry rotations — a unique bias per qubit\n",[507,85377,85378,85380,85382,85384,85386,85388,85390],{"class":509,"line":697},[507,85379,1630],{"class":513},[507,85381,8246],{"class":517},[507,85383,1636],{"class":513},[507,85385,8221],{"class":572},[507,85387,580],{"class":517},[507,85389,58245],{"class":583},[507,85391,1883],{"class":517},[507,85393,85394,85396,85398,85401,85403,85406,85408,85410,85412,85414,85417,85419,85421,85423,85425,85427,85430],{"class":509,"line":710},[507,85395,21867],{"class":517},[507,85397,639],{"class":576},[507,85399,85400],{"class":517},"((seed ",[507,85402,2391],{"class":572},[507,85404,85405],{"class":517}," (i ",[507,85407,2107],{"class":572},[507,85409,1426],{"class":583},[507,85411,655],{"class":517},[507,85413,2391],{"class":572},[507,85415,85416],{"class":583}," 0.1",[507,85418,655],{"class":517},[507,85420,8769],{"class":572},[507,85422,58644],{"class":517},[507,85424,584],{"class":583},[507,85426,8229],{"class":572},[507,85428,85429],{"class":517}," math.pi), i)",[507,85431,72708,85432],{"class":72706,"tabindex":72707},[507,85433,85434],{"class":72711,"role":72712},"A seed-based rotation biases each qubit, so every plant is unique but reproducible.",[507,85436,85437],{"class":509,"line":715},[507,85438,556],{"emptyLinePlaceholder":133},[507,85440,85441],{"class":509,"line":721},[507,85442,85443],{"class":562},"# 3 · Entanglement chain — correlate neighbouring qubits\n",[507,85445,85446,85448,85450,85452,85454,85456,85458],{"class":509,"line":736},[507,85447,1630],{"class":513},[507,85449,8246],{"class":517},[507,85451,1636],{"class":513},[507,85453,8221],{"class":572},[507,85455,580],{"class":517},[507,85457,12152],{"class":583},[507,85459,1883],{"class":517},[507,85461,85462,85464,85466,85469,85471,85473,85475],{"class":509,"line":748},[507,85463,21867],{"class":517},[507,85465,615],{"class":576},[507,85467,85468],{"class":517},"(i, i ",[507,85470,2107],{"class":572},[507,85472,1426],{"class":583},[507,85474,3649],{"class":517},[507,85476,72708,85477],{"class":72706,"tabindex":72707},[507,85478,85479,85481],{"class":72711,"role":72712},[154,85480,73431],{}," CNOTs link neighbouring qubits, so a plant's traits become correlated.",[507,85483,85484],{"class":509,"line":761},[507,85485,556],{"emptyLinePlaceholder":133},[507,85487,85488],{"class":509,"line":775},[507,85489,85490],{"class":562},"# 4 · Cross-entanglement — non-local correlations\n",[507,85492,85493,85495,85497,85499,85501,85503,85505,85508,85510,85512,85514,85516,85518,85520,85522,85524,85526,85528,85530],{"class":509,"line":784},[507,85494,593],{"class":517},[507,85496,615],{"class":576},[507,85498,580],{"class":517},[507,85500,601],{"class":583},[507,85502,622],{"class":517},[507,85504,584],{"class":583},[507,85506,85507],{"class":517},"); qc.",[507,85509,615],{"class":576},[507,85511,580],{"class":517},[507,85513,625],{"class":583},[507,85515,622],{"class":517},[507,85517,8226],{"class":583},[507,85519,85507],{"class":517},[507,85521,615],{"class":576},[507,85523,580],{"class":517},[507,85525,584],{"class":583},[507,85527,622],{"class":517},[507,85529,12152],{"class":583},[507,85531,587],{"class":517},[507,85533,85534],{"class":509,"line":796},[507,85535,556],{"emptyLinePlaceholder":133},[507,85537,85538],{"class":509,"line":809},[507,85539,85540],{"class":562},"# Measure → the plant's permanent quantum fingerprint\n",[507,85542,85543,85545,85547,85549,85551,85553,85555,85557,85559,85561,85563],{"class":509,"line":1352},[507,85544,593],{"class":517},[507,85546,72822],{"class":576},[507,85548,580],{"class":517},[507,85550,2204],{"class":572},[507,85552,580],{"class":517},[507,85554,58245],{"class":583},[507,85556,2213],{"class":517},[507,85558,2204],{"class":572},[507,85560,580],{"class":517},[507,85562,58245],{"class":583},[507,85564,22540],{"class":517},[507,85566,85567,85569,85571,85573,85575,85577,85579,85581,85583,85585,85587,85589,85591,85593],{"class":509,"line":1357},[507,85568,79916],{"class":517},[507,85570,573],{"class":572},[507,85572,73487],{"class":517},[507,85574,22501],{"class":576},[507,85576,79863],{"class":517},[507,85578,68762],{"class":2155},[507,85580,573],{"class":572},[507,85582,5682],{"class":583},[507,85584,14176],{"class":517},[507,85586,23996],{"class":576},[507,85588,13983],{"class":517},[507,85590,73558],{"class":576},[507,85592,66172],{"class":517},[507,85594,72708,85595],{"class":72706,"tabindex":72707},[507,85596,85597,85598,73508],{"class":72711,"role":72712},"Runs on real IonQ hardware through Qollab. ",[504,85599,73507],{},[72443,85601,85602],{"avatar":73990,"name":73991,"role":73981,"username":1004},[18,85603,74063],{},[18,85605,85606,85607,85611,85612,85614],{},"The rendering system went through similar problem-solving. The team started in ",[49,85608,85610],{"href":85609},"https:\u002F\u002Fpixijs.com","PixiJS",", hit performance walls, and rebuilt in ",[49,85613,73017],{"href":73016}," and WebGL, with an adaptive rendering layer that progressively scales back effects based on device capability. The result has a cosmic, outer-space quality they didn't plan for but kept.",[72443,85616,85617],{"avatar":73979,"name":73980,"role":73981,"username":1004},[18,85618,74013],{},[831,85620],{"alt":85621,"caption":85622,"no":844,"src":85623},"The Quantum Garden Seed Box: a grid catalog of 42 plant varieties, each a watercolor botanical illustration","The Seed Box: a grid catalog of 42 plant varieties, each a watercolor-style botanical illustration with its own metadata.","\u002F_content\u002Fimages\u002Fquantum-garden\u002Ffig2-seed-box.webp",[74616,85625,85627],{"lead":85626},"None of this is a black box. Quantum Garden is fully open source, end to end.",[41852,85628,85629,85637],{},[41855,85630,85631],{},[41858,85632,85633,85635],{},[41861,85634,74627],{},[41861,85636,74630],{},[41868,85638,85639,85647,85655,85663],{},[41858,85640,85641,85644],{},[41873,85642,85643],{},"Stack",[41873,85645,85646],{},"Next.js 16 · React 19 · Three.js · Qiskit · IonQ · PostgreSQL",[41858,85648,85649,85652],{},[41873,85650,85651],{},"Quantum pool",[41873,85653,85654],{},"500 authentic results, 100 from each of five circuit types: superposition, Bell pair, GHZ, interference, and the variational circuit above.",[41858,85656,85657,85660],{},[41873,85658,85659],{},"Observation",[41873,85661,85662],{},"Traits reveal in under 50 ms, chosen deterministically from each plant's ID hash. No waiting, and no two gardens alike.",[41858,85664,85665,85668],{},[41873,85666,85667],{},"Evolution",[41873,85669,85670],{},"Runs server-side. The garden germinates, blooms, and fades whether or not anyone is watching.",[13,85672,85674],{"id":85673},"what-theyd-build-next","What they'd build next",[18,85676,85677],{},"Both builders have clear ideas about where the project could go.",[18,85679,85680,85683],{},[154,85681,85682],{},"Justin"," would like to build a generative evolution system: plants combining and merging their quantum circuits to produce offspring, with circuit crossover and mutation as a metaphor for biological genetics. A quantum lineage you could trace from a two-qubit ancestor to increasingly complex descendants.",[18,85685,85686,85689],{},[154,85687,85688],{},"Amber"," wants to extend the concept into a full quantum ecosystem: weather, seasons, ambient soundscapes, and visitor traces all driven by quantum outcomes. A living world that responds to the people who return to it over time.",[18,85691,85692,85693,53],{},"Neither has done it yet. The design space is intentionally left open for others to join, contribute, and ",[49,85694,85695],{"href":85268},"add something new",[13,85697,73027],{"id":73026},[18,85699,85700],{},"You don't need a PhD to build with quantum. As Amber and Justin showed, a strong concept plus access to real hardware is enough. The fastest way in is to start from a circuit that already works.",[73026,85702,85705],{"fork-href":85268,"live-href":85703,"title":85704},"https:\u002F\u002Fwww.quantum-garden.com\u002F","Start from a circuit that already works.",[18,85706,85707,85708],{},"Fork the Quantum Garden templates and framework, and build your own living world on real hardware. ",[154,85709,73040],{},[953,85711,85712],{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":85714},[85715,85716,85717,85718,85719],{"id":85212,"depth":105,"text":85213},{"id":85236,"depth":105,"text":85237},{"id":77016,"depth":105,"text":77017},{"id":85673,"depth":105,"text":85674},{"id":73026,"depth":105,"text":73027},[112,969,1006],[85722,85730],{"name":73980,"role":85723,"avatar":73979,"bio":85724,"links":85725},"Co-founder · PressRoom AI","A data scientist, SEO strategist, and co-founder of PressRoom AI, drawn to probability, uncertainty, and complex systems across her work in commercial real-estate analytics and AI. A garden lover at heart, and proof that meaningful quantum work no longer requires a physics PhD.",[85726,85728],{"label":73068,"href":85727},"\u002Fu\u002FAmberPincar",{"label":73059,"href":85729},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Famber-wang-\u002F",{"name":73991,"role":85731,"avatar":73990,"bio":85732,"links":85733},"Co-founder & CTO · Achievable","A software engineer and CTO of Achievable whose background includes open-sourcing AdWhirl at Google and scaling it to over a billion ad impressions a day. He tracked quantum from a distance for years, treating it as out of reach, until real hardware and a strong concept changed that.",[85734],{"label":73059,"href":85735},"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fjustinpincar\u002F",{"username":8,"name":73075,"role":73076,"avatar":73077},"Amber Wang and Justin Pincar built a living digital garden where every plant gets its form and behavior from circuits run on real quantum hardware.","Qollab's Quantum Creative Challenge asked curious people to build something new with quantum computing. Amber Wang and Justin Pincar built a living digital garden powered by real quantum hardware.",{"href":85268,"label":74105},{"image":74058,"alt":85741,"liveUrl":85703},"Quantum Garden, Qollab x IonQ, Quantum Creative Challenge, Fall 2025, with builders Justin Pincar and Amber Wang",{},"\u002Fblog\u002Fquantum-garden","2026-03-25",[],[85747,85748,85749],{"username":1011,"project":1012,"title":1013,"category":1007,"thumb":1014,"to":73092},{"username":73101,"project":73102,"title":73103,"category":73097,"thumb":73104,"to":73105},{"username":74372,"project":79092,"title":74261,"category":73752,"thumb":79093,"to":74365},{"title":85751,"description":85752},"Quantum Creative Project Showcase: Quantum Garden","What if a digital garden could be powered by the actual randomness of quantum physics? Real quantum measurement outcomes, woven permanently into every plant.","blog\u002Fquantum-garden",[73112,143,73111],"KWgV_LYof1btP9J4Cn2OxC9-m3lExIlvlpH-FB6Bqf4",{"id":85757,"title":85758,"authors":85759,"body":85761,"breadcrumb":85992,"builders":85995,"byline":85996,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":85998,"description":85998,"draft":125,"extension":126,"eyebrow":116,"finish":85999,"fork":116,"hero":86000,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":86003,"navigation":133,"newsItems":116,"next":86004,"ogImage":116,"order":590,"outcomes":116,"path":86008,"publishDate":86009,"readingTime":86010,"related":86011,"relatedProjects":116,"seo":86012,"stem":86014,"tags":86015,"track":86016,"trackName":85994,"__hash__":86017},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcode-playground.md","Use the Code Playground",[85760],"qollab",{"type":10,"value":85762,"toc":85981},[85763,85773,85776,85780,85788,85796,85798,85802,85809,85829,85833,85840,85844,85851,85870,85876,85885,85889,85899,85903,85906,85930,85934,85937,85951,85959,85961,85965,85969],[18,85764,85765,85766,85768,85769,85772],{},"Every project on ",[49,85767,112],{"href":645}," comes with a built-in code playground. It is a full Python and Qiskit editor that runs your circuits right in your browser, whether you are testing on a simulator or executing on real quantum hardware from ",[49,85770,76863],{"href":85771},"https:\u002F\u002Fwww.ionq.com",". No complex installations or local environments are required!",[18,85774,85775],{},"This guide will walk you through everything you need to know to start running your code.",[13,85777,85779],{"id":85778},"prerequisites","Prerequisites",[18,85781,85782,85783,85787],{},"This guide assumes you have a free Qollab account (you can ",[49,85784,85786],{"href":85785},"\u002Flogin","sign up here"," in seconds) and are signed in.",[18,85789,85790,85791,85795],{},"To use the Playground, you need to be viewing a project page that contains Qiskit or Python code. You can experiment with the code on any public project you find on the platform. If you want to run your own custom code, check out our guide to ",[49,85792,85794],{"href":85793},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcreate-project","Create a Project on Qollab"," first.",[72127,85797],{},[13,85799,85801],{"id":85800},"step-1-open-the-playground","Step 1: Open the Playground",[18,85803,85804,85805,85808],{},"To get started, navigate to any project page. Right below the project description, you will see a ",[154,85806,85807],{},"Code Playground"," section. Review your code or tweak your project's Python script. We've built the editor to be as helpful and user-friendly as possible:",[42,85810,85811,85817,85823],{},[45,85812,85813,85816],{},[154,85814,85815],{},"Syntax Highlighting:"," Keywords, method calls, and strings are color-coded to make reading your code a breeze.",[45,85818,85819,85822],{},[154,85820,85821],{},"Smart Errors:"," If a line of code fails, the problematic line will light up red so you know exactly where to look.",[45,85824,85825,85828],{},[154,85826,85827],{},"Focus Mode:"," Click the fullscreen icon in the toolbar when you need a distraction-free coding environment.",[13,85830,85832],{"id":85831},"step-2-edit-your-code","Step 2: Edit Your Code",[18,85834,85835,85836,85839],{},"Click on ",[154,85837,85838],{},"Run"," to switch to the coderunner. You will see a dropdown menu, a Run button, and a dark console panel waiting for your commands. Think of this as your quantum command center!",[13,85841,85843],{"id":85842},"step-3-select-a-qpu","Step 3: Select a QPU",[18,85845,85846,85847,85850],{},"Use the ",[154,85848,85849],{},"Select QPU"," dropdown to choose where your circuit will execute:",[42,85852,85853,85859],{},[45,85854,85855,85858],{},[154,85856,85857],{},"Simulators:"," These are free, run instantly, and are always available. They are the perfect sandbox for testing your logic and catching errors without using any resources.",[45,85860,85861,85864,85865,85869],{},[154,85862,85863],{},"Hardware:"," Ready for the real thing? You can select physical ",[49,85866,85868],{"href":85867},"https:\u002F\u002Fwww.ionq.com\u002Fquantum-systems\u002Fcompare","IonQ quantum computers"," (like Harmony, Aria, or Forte).",[18,85871,85872,85875],{},[154,85873,85874],{},"Managing Credits:"," Running on actual hardware consumes credits. You can view your remaining balance or set a spending limit directly in your Qollab Profile Settings to ensure you always stay within your budget.",[85877,85878,85879],"aside-note",{},[18,85880,85881,85884],{},[154,85882,85883],{},"Pro tip:"," Use simulators for your initial runs to catch syntax errors or logic bugs. This ensures that when you do use your hardware credits, you're running a polished, verified circuit.",[13,85886,85888],{"id":85887},"step-4-run-your-circuit","Step 4: Run Your Circuit",[18,85890,85891,85892,85894,85895,85898],{},"Click the ",[154,85893,85838],{}," button to execute your code. The console background will turn blue, and a timer will start tracking the elapsed time. If you ever need to stop the execution, an ",[154,85896,85897],{},"Interrupt"," button will appear while it's processing.",[13,85900,85902],{"id":85901},"step-5-read-the-output","Step 5: Read the Output",[18,85904,85905],{},"Once your run is complete, the results will populate directly in the console panel below. Here is what you can expect to see:",[42,85907,85908,85918,85924],{},[45,85909,85910,85913,85914,85917],{},[154,85911,85912],{},"Standard Output:"," Any ",[504,85915,85916],{},"print()"," statements you included in your code.",[45,85919,85920,85923],{},[154,85921,85922],{},"Errors:"," If something went wrong, you'll see a clear, red traceback pointing to the exact issue.",[45,85925,85926,85929],{},[154,85927,85928],{},"Circuit & Probabilities:"," After a successful run, you will often see a visual diagram of the circuit that was sent to the QPU, along with a bar chart showing the measurement probabilities of your outcomes.",[13,85931,85933],{"id":85932},"step-6-copy-or-download","Step 6: Copy or Download",[18,85935,85936],{},"Need to save your results or share them in your project documentation? The toolbar just above the console has you covered:",[42,85938,85939,85945],{},[45,85940,85941,85944],{},[154,85942,85943],{},"Copy:"," Quickly copy all the console text to your clipboard.",[45,85946,85947,85950],{},[154,85948,85949],{},"Download:"," Save the full output as an HTML file, complete with your project name and a timestamp.",[85877,85952,85953],{},[18,85954,85955,85958],{},[154,85956,85957],{},"Community impact:"," Sharing your actual hardware run results in your project's Markdown documentation is incredibly helpful! It allows other learners to see real-world quantum noise and performance data without having to spend their own credits.",[72127,85960],{},[13,85962,85964],{"id":85963},"technical-requirements","Technical Requirements",[13,85966,85968],{"id":85967},"browser-compatibility","Browser Compatibility",[18,85970,85971,85972,85975,85976,56374,85979,14176],{},"To ensure the Code Playground runs smoothly, we recommend using a modern browser (like Chrome, Edge, or Opera) that supports WebAssembly JSPI. If you see a compatibility warning, simply switch to Chrome, Edge, or Opera. (If you are a Firefox user, you can enable this manually by going to ",[504,85973,85974],{},"about:config"," and setting ",[504,85977,85978],{},"javascript.options.wasm_js_promise_integration",[504,85980,2557],{},{"title":104,"searchDepth":105,"depth":105,"links":85982},[85983,85984,85985,85986,85987,85988,85989,85990,85991],{"id":85778,"depth":105,"text":85779},{"id":85800,"depth":105,"text":85801},{"id":85831,"depth":105,"text":85832},{"id":85842,"depth":105,"text":85843},{"id":85887,"depth":105,"text":85888},{"id":85901,"depth":105,"text":85902},{"id":85932,"depth":105,"text":85933},{"id":85963,"depth":105,"text":85964},{"id":85967,"depth":105,"text":85968},[112,85993,85994,85758],"Learn","Building your first Qollab project",[],{"username":85760,"name":112,"role":85997,"avatar":104},"Product docs","Run Qiskit circuits live in your browser, from editing Python to choosing a QPU and reading your results.","That’s the course. Your first project runs on real quantum hardware.",{"image":86001,"alt":85758},"\u002F_content\u002Fimages\u002Fcode-playground\u002Fhero.webp","lesson",{},{"slug":86005,"title":86006,"desc":86007},"\u002Fprojects","Share what you built","Publish your project page so others can fork it, and browse what the community is building.","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcode-playground","2026-03-06","3 min read",[],{"title":86013,"description":85998},"Use the Code Playground · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcode-playground",[],"building-your-first-qollab-project","AP5bvOyLnFqXa42pAO5TPnI0xsntFECJ9XUKSDd-1SU",{"id":86019,"title":85794,"authors":86020,"body":86021,"breadcrumb":86342,"builders":86343,"byline":86344,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":86345,"description":86346,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":86347,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":86349,"navigation":133,"newsItems":116,"next":86350,"ogImage":116,"order":566,"outcomes":116,"path":86352,"publishDate":86009,"readingTime":72277,"related":86353,"relatedProjects":116,"seo":86354,"stem":86356,"tags":86357,"track":86016,"trackName":85994,"__hash__":86358},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcreate-project.md",[85760],{"type":10,"value":86022,"toc":86334},[86023,86030,86034,86049,86053,86056,86058,86062,86068,86072,86115,86119,86129,86133,86137,86140,86144,86147,86253,86260,86264,86267,86271,86285,86292,86296,86299,86303,86321,86328,86331],[18,86024,86025,86026,86029],{},"This guide assumes that you have already created a free Qollab account, chosen a username, and are currently signed into the platform. If any of that sounds unfamiliar, don't fret, ",[49,86027,86028],{"href":85785},"sign in or sign up here"," first, then come back to start flexing your quantum muscles.",[13,86031,86033],{"id":86032},"navigating-to-the-creation-page","Navigating to the Creation Page",[18,86035,86036,86037,86040,86041,86044,86045,53],{},"To start a new project, simply click the ",[1031,86038,86039],{},"Add Project"," button on the ",[49,86042,86043],{"href":86005},"Projects"," page or directly create a new ",[49,86046,86048],{"href":86047},"\u002Fnew","project",[831,86050],{"alt":86051,"caption":104,"no":104,"src":86052},"The Projects page showing the Add Project button in the top right corner","\u002F_content\u002Fimages\u002Fcreate-project\u002Fqollab-projects-add-button-1773201441250.webp",[18,86054,86055],{},"You will enter a five-step creation process. Each step is saved automatically as you progress, so you won't lose your work if you happen to navigate away.",[72127,86057],{},[13,86059,86061],{"id":86060},"step-1-project-details","Step 1: Project Details",[18,86063,86064,86065],{},"The first step collects the essential information about your project. ",[1031,86066,86067],{},"(Note: Project Title and Description are required fields.)",[831,86069],{"alt":86070,"caption":104,"no":104,"src":86071},"Step 1, Project Details form","\u002F_content\u002Fimages\u002Fcreate-project\u002Fstep1-project-details-png-1773201592099.webp",[42,86073,86074,86083,86093,86099],{},[45,86075,86076,86079,86080,14176],{},[154,86077,86078],{},"Project Title:"," Enter a concise but descriptive name (e.g., ",[1031,86081,86082],{},"Quantum Teleportation Visualizer",[45,86084,86085,86088,86089,86092],{},[154,86086,86087],{},"Project URL Name:"," As you type your title, Qollab automatically generates a web address for your project (e.g., ",[504,86090,86091],{},"quantum-teleportation-visualizer","). You can manually edit this if you want to make it shorter.",[45,86094,86095,86098],{},[154,86096,86097],{},"Description:"," Write a brief summary of what your project does (up to 1,024 characters). This acts as a preview card when people are browsing.",[45,86100,86101,86104,86105,622,86108,86110,86111,86114],{},[154,86102,86103],{},"Labels (optional):"," Add tags like ",[504,86106,86107],{},"qiskit",[504,86109,142],{},", or ",[504,86112,86113],{},"entanglement",". Clear details and accurate labels make it easy for the community to discover your work when searching for specific quantum topics.",[13,86116,86118],{"id":86117},"step-2-project-content","Step 2: Project Content",[18,86120,86121,86122,10799,86125,86128],{},"This required field is where you explain the ",[1031,86123,86124],{},"why",[1031,86126,86127],{},"how"," behind your project, documenting your thought process, your goals, and your results, helping others learn from your experience. The editor on this field supports standard Markdown syntax for formatting text.",[831,86130],{"alt":86131,"caption":104,"no":104,"src":86132},"Step 2, Project Content markdown editor","\u002F_content\u002Fimages\u002Fcreate-project\u002Fstep2-content-png-1773201624099.webp",[13,86134,86136],{"id":86135},"step-3-project-code","Step 3: Project Code",[18,86138,86139],{},"This required step is where you add your quantum computing scripts. Qollab pre-loads the editor with a simple Python starter script using Qiskit (a popular quantum SDK), a two-qubit Bell state circuit, just to give you a foundation to build from:",[831,86141],{"alt":86142,"caption":104,"no":104,"src":86143},"Step 3, Project Code editor with Python\u002FQiskit starter template","\u002F_content\u002Fimages\u002Fcreate-project\u002Fstep3-code-png-1773201641617.webp",[18,86145,86146],{},"You can replace this entirely with your own code or build directly upon it.",[498,86148,86150],{"className":500,"code":86149,"language":502,"meta":104,"style":104},"from qiskit import QuantumCircuit\nfrom qiskit.quantum_info import Statevector\n\n# Create a 2-qubit circuit\nqc = QuantumCircuit(2)\n\n# Apply gates\nqc.h(0)  # Hadamard gate on qubit 0\nqc.cx(0, 1)  # CNOT gate (creates entanglement)\n\n# Display the circuit\nprint(qc)\n",[504,86151,86152,86162,86172,86176,86181,86195,86199,86204,86219,86238,86242,86247],{"__ignoreMap":104},[507,86153,86154,86156,86158,86160],{"class":509,"line":510},[507,86155,529],{"class":513},[507,86157,532],{"class":517},[507,86159,514],{"class":513},[507,86161,537],{"class":517},[507,86163,86164,86166,86168,86170],{"class":509,"line":105},[507,86165,529],{"class":513},[507,86167,545],{"class":517},[507,86169,514],{"class":513},[507,86171,84694],{"class":517},[507,86173,86174],{"class":509,"line":540},[507,86175,556],{"emptyLinePlaceholder":133},[507,86177,86178],{"class":509,"line":553},[507,86179,86180],{"class":562},"# Create a 2-qubit circuit\n",[507,86182,86183,86185,86187,86189,86191,86193],{"class":509,"line":559},[507,86184,569],{"class":517},[507,86186,573],{"class":572},[507,86188,577],{"class":576},[507,86190,580],{"class":517},[507,86192,584],{"class":583},[507,86194,587],{"class":517},[507,86196,86197],{"class":509,"line":566},[507,86198,556],{"emptyLinePlaceholder":133},[507,86200,86201],{"class":509,"line":590},[507,86202,86203],{"class":562},"# Apply gates\n",[507,86205,86206,86208,86210,86212,86214,86216],{"class":509,"line":610},[507,86207,593],{"class":517},[507,86209,596],{"class":576},[507,86211,580],{"class":517},[507,86213,601],{"class":583},[507,86215,22718],{"class":517},[507,86217,86218],{"class":562},"# Hadamard gate on qubit 0\n",[507,86220,86221,86223,86225,86227,86229,86231,86233,86235],{"class":509,"line":634},[507,86222,593],{"class":517},[507,86224,615],{"class":576},[507,86226,580],{"class":517},[507,86228,601],{"class":583},[507,86230,622],{"class":517},[507,86232,625],{"class":583},[507,86234,22718],{"class":517},[507,86236,86237],{"class":562},"# CNOT gate (creates entanglement)\n",[507,86239,86240],{"class":509,"line":661},[507,86241,556],{"emptyLinePlaceholder":133},[507,86243,86244],{"class":509,"line":678},[507,86245,86246],{"class":562},"# Display the circuit\n",[507,86248,86249,86251],{"class":509,"line":683},[507,86250,8525],{"class":572},[507,86252,694],{"class":517},[85877,86254,86255],{},[18,86256,86257,86259],{},[154,86258,85957],{}," Sharing your actual code gives the community a tangible, runnable example to learn from. Others can test your logic, discover new implementation techniques, or even build upon your foundation to create something entirely new.",[13,86261,86263],{"id":86262},"step-4-project-links","Step 4: Project Links",[18,86265,86266],{},"If you have files or websites related to your project hosted somewhere else, you can link them here:",[831,86268],{"alt":86269,"caption":104,"no":104,"src":86270},"Step 4, Project Links with Git Repository and Project Website fields","\u002F_content\u002Fimages\u002Fcreate-project\u002Fstep4-links-png-1773201658327.webp",[42,86272,86273,86279],{},[45,86274,86275,86278],{},[154,86276,86277],{},"Git Repository:"," Provide a link to your source code on GitHub, GitLab, or Bitbucket.",[45,86280,86281,86284],{},[154,86282,86283],{},"Project Website:"," Provide a link to a live demo, a Jupyter notebook, or an arXiv paper.",[85877,86286,86287],{},[18,86288,86289,86291],{},[154,86290,85957],{}," Linking to external repositories or live demos gives the community a much deeper look into your project. It opens the door for others to actively contribute to your codebase, review your full research, or see your work functioning in the real world.",[13,86293,86295],{"id":86294},"step-5-publish","Step 5: Publish",[18,86297,86298],{},"Congratulations, your project is saved! You now have two choices:",[831,86300],{"alt":86301,"caption":104,"no":104,"src":86302},"Step 5, Publish options with Publish Now and Keep as Draft","\u002F_content\u002Fimages\u002Fcreate-project\u002Fstep5-publish-png-1773201675400.webp",[42,86304,86305,86315],{},[45,86306,86307,86310,86311,86314],{},[154,86308,86309],{},"Publish Now:"," This makes your project publicly visible to the community and tags it as your official ",[1031,86312,86313],{},"v1.0"," release. Publishing requires you to agree to the Community Guidelines.",[45,86316,86317,86320],{},[154,86318,86319],{},"Keep as Draft:"," This saves your project privately. You can continue updating your draft and publish it whenever you are completely ready.",[85877,86322,86323],{},[18,86324,86325,86327],{},[154,86326,85957],{}," Publishing is what makes open collaboration possible! By sharing your work publicly, you invite constructive feedback, spark interesting discussions, and actively contribute to the growing, shared knowledge base of the quantum computing community.",[18,86329,86330],{},"We look forward to seeing what you will build and share with the community!",[953,86332,86333],{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":86335},[86336,86337,86338,86339,86340,86341],{"id":86032,"depth":105,"text":86033},{"id":86060,"depth":105,"text":86061},{"id":86117,"depth":105,"text":86118},{"id":86135,"depth":105,"text":86136},{"id":86262,"depth":105,"text":86263},{"id":86294,"depth":105,"text":86295},[112,85993,85994,85794],[],{"username":85760,"name":112,"role":85997,"avatar":104},"This guide highlights the steps to build and publish your project; a step-by-step walkthrough of Qollab's project creation process, from naming your work to releasing it for the community to discover.","A step-by-step walkthrough of Qollab's project creation, from naming your work to releasing it for the community to discover.",{"image":86348,"alt":85794},"\u002F_content\u002Fimages\u002Fcreate-project\u002Fhero.webp",{},{"slug":86008,"title":86351,"desc":85998},"7 · Use the Code Playground","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcreate-project",[],{"title":86355,"description":86346},"Create a Project on Qollab · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fcreate-project",[],"MIg53Bk8sB6OIQcy9_ioz1Omvlry8U5LTuh49hSsjFk",{"id":86360,"title":86361,"authors":86362,"body":86364,"breadcrumb":87127,"builders":87128,"byline":87129,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":87132,"description":87133,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":87134,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":87136,"navigation":133,"newsItems":116,"next":87137,"ogImage":116,"order":559,"outcomes":116,"path":87140,"publishDate":87141,"readingTime":74132,"related":87142,"relatedProjects":116,"seo":87143,"stem":87145,"tags":87146,"track":86016,"trackName":85994,"__hash__":87147},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-hardware.md","Run circuits on IonQ’s quantum hardware",[86363],"stewart-smith",{"type":10,"value":86365,"toc":87113},[86366,86422,86426,86436,86457,86480,86484,86487,86491,86526,86530,86535,86537,86541,86556,86792,86809,86813,86821,86824,86828,86846,86850,86857,86908,86914,86918,86928,86932,86939,86942,86955,87056,87059,87099,87103,87110],[18,86367,86368,86371,86372,86376,86377,86381,86382,622,86386,622,86390,622,86394,622,86398,86402,86403,86407,86408,86412,86413,86417,86418,53],{},[49,86369,76863],{"href":86370},"https:\u002F\u002Fwww.ionq.com\u002F"," is a leading quantum hardware startup, developing general-purpose ",[49,86373,86375],{"href":86374},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTrapped-ion_quantum_computer","trapped ion quantum computers"," and accompanying software to generate, optimize, and execute ",[49,86378,86380],{"href":86379},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_circuit","quantum circuits",". In this guide we will ",[49,86383,86385],{"href":86384},"#_1-upgrade-your-ionq-account","upgrade your IonQ account to a paid tier",[49,86387,86389],{"href":86388},"#_2-prepare-our-ionq-project","prepare our IonQ project",[49,86391,86393],{"href":86392},"#_3-confirm-your-ionq-qpu-access","confirm our QPU access",[49,86395,86397],{"href":86396},"#_5-estimate-your-ionq-spend","estimate our IonQ spend",[49,86399,86401],{"href":86400},"#_6-submit-your-circuit-to-ionqs-quantum-hardware-queue","send our quantum circuit to IonQ’s quantum hardware",", and ultimately ",[49,86404,86406],{"href":86405},"#_7-retrieve-ionq-circuit-results-asynchronously","retrieve our quantum circuit’s results asynchronously",". In order to follow along, it is essential that you have completed our previous guides for ",[49,86409,86411],{"href":86410},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-setup","installing Python via UV"," (which also covers operating a shell command-line interface), and ",[49,86414,86416],{"href":86415},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup","setting up an IonQ project",". For additional information on IonQ’s SDK, see ",[49,86419,86421],{"href":86420},"https:\u002F\u002Fdocs.ionq.com\u002Fsdks\u002Fqiskit","IonQ’s Qiskit tutorial",[13,86423,86425],{"id":86424},"_1-upgrade-your-ionq-account","1. Upgrade your IonQ account",[18,86427,86428,86429,86431,86432,86435],{},"It’s important to understand that the use of ",[1031,86430,59914],{}," quantum hardware (versus ",[1031,86433,86434],{},"simulated"," quantum hardware) is inherently expensive. Completing this tutorial will require you to purchase compute power from IonQ.",[18,86437,86438,86439,86442,86443,622,86447,86451,86452,86456],{},"In our ",[49,86440,86441],{"href":86415},"previous IonQ guide"," we have ",[49,86444,86446],{"href":86445},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#_1-create-an-ionq-account","created an IonQ account",[49,86448,86450],{"href":86449},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#_2-generate-an-ionq-api-key","generated an IonQ API key",", and put it to use running a quantum circuit on ",[49,86453,86455],{"href":86454},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#_6-run-your-circuit-on-ionqs-cloud-simulator","IonQ’s quantum simulator",". Fantastic. But this is as far as we can go using IonQ’s free account tier. In order to run our quantum circuit on actual IonQ hardware, you must upgrade to a paid IonQ account.",[18,86458,86459,86460,86464,86465,86469,86470,86474,86475,86479],{},"If you’ve submitted a ",[49,86461,86463],{"href":86462},"\u002Frfp","Qollab RFP"," and have been approved for a grant, your IonQ account will include a generous number of credits that can be used towards a compute spend. If you do not have available IonQ credits, visit the ",[49,86466,86468],{"href":86467},"https:\u002F\u002Fcloud.ionq.com\u002Fbackends","“Backends” page of your IonQ account",", select a QPU that you would like to use, and click its “Out of Plan, Request Access” badge. Follow the prompts to request access and join a paid account tier. Additionally, you have the option to email ",[49,86471,86473],{"href":86472},"mailto:support@ionq.com","support@ionq.com"," or fill out ",[49,86476,86478],{"href":86477},"https:\u002F\u002Fsupport.ionq.com\u002Fhc\u002Fen-us\u002Frequests\u002Fnew?","this support form"," to discuss account options.",[86481,86482],"cta-link",{"href":86467,"label":86483},"IonQ Backends →",[18,86485,86486],{},"This first step in our guide to running circuits on IonQ’s quantum hardware will take some time because it requires dialogue with real human beings. Be patient. Good things will come.",[13,86488,86490],{"id":86489},"_2-prepare-our-ionq-project","2. Prepare our IonQ project",[18,86492,86438,86493,86495,86496,622,86500,622,86504,86508,86509,86513,86514,86517,86518,86521,86522,86525],{},[49,86494,86441],{"href":86415}," we ",[49,86497,86499],{"href":86498},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#_3-create-a-new-project","created a new project",[49,86501,86503],{"href":86502},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#install-ionqs-sdk","installed IonQ’s SDK",[49,86505,86507],{"href":86506},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#_5-update-our-transpile-settings","updated our transpile settings",", and ",[49,86510,86512],{"href":86511},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup#include-your-key-in-our-project","safely included our IonQ API key inside our project",". If you skipped these steps, now is the perfect time to tend to them as we will build off of their output here. You will also need to ",[49,86515,86516],{"href":86449},"generate an IonQ API key"," if you have not done so already. Once you have ",[49,86519,86520],{"href":86384},"upgraded your IonQ account to a paid tier"," (see above) and prepared your IonQ Python project (as described in our ",[49,86523,86524],{"href":86415},"previous guide","), you will be ready to proceed to the next step.",[13,86527,86529],{"id":86528},"_3-confirm-your-ionq-qpu-access","3. Confirm your IonQ QPU access",[18,86531,86532,86533,53],{},"Your IonQ account has been upgraded, and your local Python project is setup for action. Let’s confirm what IonQ hardware your API key can access. Point your Web browser to the ",[49,86534,86468],{"href":86467},[86481,86536],{"href":86467,"label":86483},[13,86538,86540],{"id":86539},"_4-prepare-our-quantum-circuit","4. Prepare our quantum circuit",[18,86542,86543,86544,86547,86548,86551,86552,86555],{},"Create a new blank file within your project folder, name it ",[504,86545,86546],{},"ionq-hardware-submit.py",", open it with your source code editor, and paste the following code into it. It should look very similar to our ",[504,86549,86550],{},"ionq-simulator.py"," code from our ",[49,86553,86554],{"href":86415},"previous tutorial",". Some differences have been highlighted here:",[498,86557,86559],{"className":500,"code":86558,"language":502,"meta":104,"style":104},"from qiskit import QuantumCircuit\nfrom qiskit_ionq import IonQProvider, ErrorMitigation\n\nqc = QuantumCircuit( 2, name=\"Bell state on IonQ QPU\" )\nqc.h( 0 )\nqc.cx( 0, 1 )\nqc.measure_all()\n\nprovider = IonQProvider()\nbackend = provider.get_backend( \"qpu.aria-1\" )\njob = backend.run( \n    qc, \n    shots=1000,\n    error_mitigation=ErrorMitigation.NO_DEBIASING\n)\nprint( \"Submitted job:\", job.job_id() )\nprint( \"Status:\", job.status() )\n",[504,86560,86561,86571,86589,86593,86625,86637,86653,86662,86666,86677,86705,86718,86723,86734,86752,86756,86776],{"__ignoreMap":104},[507,86562,86563,86565,86567,86569],{"class":509,"line":510},[507,86564,529],{"class":513},[507,86566,532],{"class":517},[507,86568,514],{"class":513},[507,86570,537],{"class":517},[507,86572,86573,86575,86578,86580,86583],{"class":509,"line":105},[507,86574,529],{"class":513},[507,86576,86577],{"class":517}," qiskit_ionq ",[507,86579,514],{"class":513},[507,86581,86582],{"class":517}," IonQProvider",[86584,86585,86586],"mark",{},[507,86587,86588],{"class":517},", ErrorMitigation",[507,86590,86591],{"class":509,"line":540},[507,86592,556],{"emptyLinePlaceholder":133},[507,86594,86595,86597,86599,86601,86604,86606,86608,86610,86612,86615,86620,86622],{"class":509,"line":553},[507,86596,569],{"class":517},[507,86598,573],{"class":572},[507,86600,577],{"class":576},[507,86602,86603],{"class":517},"( ",[507,86605,584],{"class":583},[507,86607,622],{"class":517},[507,86609,22008],{"class":2155},[507,86611,573],{"class":572},[507,86613,86614],{"class":730},"\"Bell state on ",[86584,86616,86617],{},[507,86618,86619],{"class":730},"IonQ QPU",[507,86621,22281],{"class":730},[507,86623,86624],{"class":517}," )\n",[507,86626,86627,86629,86631,86633,86635],{"class":509,"line":559},[507,86628,593],{"class":517},[507,86630,596],{"class":576},[507,86632,86603],{"class":517},[507,86634,601],{"class":583},[507,86636,86624],{"class":517},[507,86638,86639,86641,86643,86645,86647,86649,86651],{"class":509,"line":566},[507,86640,593],{"class":517},[507,86642,615],{"class":576},[507,86644,86603],{"class":517},[507,86646,601],{"class":583},[507,86648,622],{"class":517},[507,86650,625],{"class":583},[507,86652,86624],{"class":517},[507,86654,86655,86657,86660],{"class":509,"line":590},[507,86656,593],{"class":517},[507,86658,86659],{"class":576},"measure_all",[507,86661,781],{"class":517},[507,86663,86664],{"class":509,"line":610},[507,86665,556],{"emptyLinePlaceholder":133},[507,86667,86668,86671,86673,86675],{"class":509,"line":634},[507,86669,86670],{"class":517},"provider ",[507,86672,573],{"class":572},[507,86674,86582],{"class":576},[507,86676,781],{"class":517},[507,86678,86679,86681,86683,86686,86689,86691,86693,86701,86703],{"class":509,"line":661},[507,86680,21937],{"class":517},[507,86682,573],{"class":572},[507,86684,86685],{"class":517}," provider.",[507,86687,86688],{"class":576},"get_backend",[507,86690,86603],{"class":517},[507,86692,22281],{"class":730},[86584,86694,86695],{},[86696,86697,86698],"var",{},[507,86699,86700],{"class":730},"qpu.aria-1",[507,86702,22281],{"class":730},[507,86704,86624],{"class":517},[507,86706,86707,86709,86711,86713,86715],{"class":509,"line":678},[507,86708,23964],{"class":517},[507,86710,573],{"class":572},[507,86712,73487],{"class":517},[507,86714,22501],{"class":576},[507,86716,86717],{"class":517},"( \n",[507,86719,86720],{"class":509,"line":683},[507,86721,86722],{"class":517},"    qc, \n",[507,86724,86725,86728,86730,86732],{"class":509,"line":697},[507,86726,86727],{"class":2155},"    shots",[507,86729,573],{"class":572},[507,86731,79870],{"class":583},[507,86733,1409],{"class":517},[507,86735,86736,86739],{"class":509,"line":710},[507,86737,86738],{"class":2155},"    ",[86584,86740,86741,86744,86746,86749],{},[507,86742,86743],{"class":2155},"error_mitigation",[507,86745,573],{"class":572},[507,86747,86748],{"class":517},"ErrorMitigation.",[507,86750,86751],{"class":583},"NO_DEBIASING",[507,86753,86754],{"class":509,"line":715},[507,86755,587],{"class":517},[507,86757,86758],{"class":509,"line":721},[86584,86759,86760,86762,86764,86767,86770,86773],{},[507,86761,8525],{"class":572},[507,86763,86603],{"class":517},[507,86765,86766],{"class":730},"\"Submitted job:\"",[507,86768,86769],{"class":517},", job.",[507,86771,86772],{"class":576},"job_id",[507,86774,86775],{"class":517},"() )",[507,86777,86778,86780,86782,86785,86787,86789],{"class":509,"line":736},[507,86779,8525],{"class":572},[507,86781,86603],{"class":517},[507,86783,86784],{"class":730},"\"Status:\"",[507,86786,86769],{"class":517},[507,86788,73518],{"class":576},[507,86790,86791],{"class":517},"() )\n",[18,86793,86794,86795,86798,86799,86801,86802,86804,86805,86808],{},"Be certain to swap out the QPU ",[504,86796,86797],{},"id"," in the example above for the ",[504,86800,86797],{}," of a QPU that you have access to. For example, our code above calls upon the ",[504,86803,86700],{}," QPU, but perhaps you are using ",[504,86806,86807],{},"qpu.forte-1",", etc.",[13,86810,86812],{"id":86811},"_5-estimate-your-ionq-spend","5. Estimate your IonQ spend",[18,86814,86815,86816,86820],{},"How much credit might you need for experimenting? ",[49,86817,86819],{"href":86818},"https:\u002F\u002Fwww.ionq.com\u002Fprograms\u002Fresearch-credits\u002Fresource-estimator","IonQ’s Resource Estimator"," enables you to predict your potential spend ahead of time according to your circuit’s number of qubit registers, gates, and so on. Our simple Bell state circuit above, run for one thousand shots, one single time, on IonQ’s “Aria” device without error mitigation, would cost just a bit above USD 12 at current early 2026 rates. But the same setup run on IonQ’s “Forte” architecture with error mitigation enabled would cost nearly USD 170. (You’ll find that the cost of error mitigation is relatively higher for small to medium circuits, and relatively lower for large circuits.) Always use the IonQ’s Resource Estimator beforehand to reduce unwanted surprises.",[86481,86822],{"href":86818,"label":86823},"IonQ Resource Estimator →",[2513,86825,86827],{"id":86826},"a-note-on-error-mitigation","A note on error mitigation",[18,86829,86830,86831,86833,86834,86837,86838,86842,86843,86845],{},"IonQ enables error mitigation be default. In our ",[504,86832,86546],{}," script above, we purposely disable error mitigation. Why would we do this? Because error mitigation is rather expensive and we don’t need it in order to demonstrate how to send jobs to IonQ’s QPUs. However, it’s likely that ",[1031,86835,86836],{},"you will want to enable error mitigation on future jobs"," in order to obtain the most useful results from real quantum hardware. (What is IonQ’s quantum error mitigation, and how does it work? Read ",[49,86839,86841],{"href":86840},"https:\u002F\u002Fdocs.ionq.com\u002Fguides\u002Ferror-mitigation-debiasing","IonQ’s guide to debiasing"," for more information.) Experiment with ",[49,86844,86819],{"href":86818}," and use your own judgement regarding when to employ error mitigation for your own circuits.",[13,86847,86849],{"id":86848},"_6-submit-your-circuit-to-ionqs-quantum-hardware-queue","6. Submit your circuit to IonQ’s quantum hardware queue",[18,86851,86852,86853,86856],{},"It’s time. Everything has led up to this moment of executing your quantum circuit on ",[1031,86854,86855],{},"real quantum hardware."," Enter the following into our shell:",[498,86858,86862],{"className":86859,"code":86860,"language":86861,"meta":104,"style":104},"language-bash shiki shiki-themes one-dark-pro","set -a;\nsource .\u002F.env; \nset +a; \nuv run python ionq-hardware-submit.py\n","bash",[504,86863,86864,86874,86885,86894],{"__ignoreMap":104},[507,86865,86866,86868,86871],{"class":509,"line":510},[507,86867,1944],{"class":572},[507,86869,86870],{"class":583}," -a",[507,86872,86873],{"class":517},";\n",[507,86875,86876,86879,86882],{"class":509,"line":105},[507,86877,86878],{"class":572},"source",[507,86880,86881],{"class":730}," .\u002F.env",[507,86883,86884],{"class":517},"; \n",[507,86886,86887,86889,86892],{"class":509,"line":540},[507,86888,1944],{"class":572},[507,86890,86891],{"class":730}," +a",[507,86893,86884],{"class":517},[507,86895,86896,86899,86902,86905],{"class":509,"line":553},[507,86897,86898],{"class":576},"uv",[507,86900,86901],{"class":730}," run",[507,86903,86904],{"class":730}," python",[507,86906,86907],{"class":730}," ionq-hardware-submit.py\n",[18,86909,86910,86911,86913],{},"Our shell will respond with a job ",[504,86912,86797],{},". This is your ticket to read the results of your real quantum operation once those results are ready.",[2513,86915,86917],{"id":86916},"patience-please","Patience, please",[18,86919,86920,86921,86924,86925,53],{},"While simulator results arrive near-instantly (at least for simple circuits), ",[154,86922,86923],{},"hardware jobs are queued, processed in order of their queue index, and take some time to resolve",". To get an overview of current IonQ QPU job queue times, visit your account’s ",[49,86926,86927],{"href":86467},"Backend page",[13,86929,86931],{"id":86930},"_7-retrieve-ionq-circuit-results-asynchronously","7. Retrieve IonQ circuit results asynchronously",[18,86933,86934,86935,53],{},"Once we’ve submitted a job to one of IonQ’s QPUs, we can check on that job’s status using the ",[49,86936,86938],{"href":86937},"https:\u002F\u002Fcloud.ionq.com\u002Fjobs","“My Jobs” tab of your account page",[86481,86940],{"href":86937,"label":86941},"IonQ “My jobs” →",[18,86943,86944,86945,86948,86949,86952,86953,53],{},"You can also programmatically poll the status of a job using a script similar to the following. Create a new file within your project’s folder and name it ",[504,86946,86947],{},"ionq-hardware-results.py",". Copy the following code into your file, replacing “",[504,86950,86951],{},"PASTE_JOB_ID_HERE","” with your real job’s ",[504,86954,86797],{},[498,86956,86958],{"className":500,"code":86957,"language":502,"meta":104,"style":104},"from qiskit_ionq import IonQProvider\n\nprovider = IonQProvider()\nbackend  = provider.get_backend( \"qpu.aria-1\" )\njob = backend.retrieve_job( \"PASTE_JOB_ID_HERE\" )\nprint( job.status() )\nprint( job.get_counts() )\n",[504,86959,86960,86971,86975,86985,87010,87035,87046],{"__ignoreMap":104},[507,86961,86962,86964,86966,86968],{"class":509,"line":510},[507,86963,529],{"class":513},[507,86965,86577],{"class":517},[507,86967,514],{"class":513},[507,86969,86970],{"class":517}," IonQProvider\n",[507,86972,86973],{"class":509,"line":105},[507,86974,556],{"emptyLinePlaceholder":133},[507,86976,86977,86979,86981,86983],{"class":509,"line":540},[507,86978,86670],{"class":517},[507,86980,573],{"class":572},[507,86982,86582],{"class":576},[507,86984,781],{"class":517},[507,86986,86987,86990,86992,86994,86996,86998,87000,87006,87008],{"class":509,"line":553},[507,86988,86989],{"class":517},"backend  ",[507,86991,573],{"class":572},[507,86993,86685],{"class":517},[507,86995,86688],{"class":576},[507,86997,86603],{"class":517},[507,86999,22281],{"class":730},[86584,87001,87002],{},[86696,87003,87004],{},[507,87005,86700],{"class":730},[507,87007,22281],{"class":730},[507,87009,86624],{"class":517},[507,87011,87012,87014,87016,87018,87021,87023,87025,87031,87033],{"class":509,"line":559},[507,87013,23964],{"class":517},[507,87015,573],{"class":572},[507,87017,73487],{"class":517},[507,87019,87020],{"class":576},"retrieve_job",[507,87022,86603],{"class":517},[507,87024,22281],{"class":730},[86584,87026,87027],{},[86696,87028,87029],{},[507,87030,86951],{"class":730},[507,87032,22281],{"class":730},[507,87034,86624],{"class":517},[507,87036,87037,87039,87042,87044],{"class":509,"line":566},[507,87038,8525],{"class":572},[507,87040,87041],{"class":517},"( job.",[507,87043,73518],{"class":576},[507,87045,86791],{"class":517},[507,87047,87048,87050,87052,87054],{"class":509,"line":590},[507,87049,8525],{"class":572},[507,87051,87041],{"class":517},[507,87053,73558],{"class":576},[507,87055,86791],{"class":517},[18,87057,87058],{},"Enter the following into your shell, from within your project’s folder:",[498,87060,87062],{"className":86859,"code":87061,"language":86861,"meta":104,"style":104},"set -a; \nsource .\u002F.env; \nset +a; \nuv run python ionq-hardware-results.py\n",[504,87063,87064,87072,87080,87088],{"__ignoreMap":104},[507,87065,87066,87068,87070],{"class":509,"line":510},[507,87067,1944],{"class":572},[507,87069,86870],{"class":583},[507,87071,86884],{"class":517},[507,87073,87074,87076,87078],{"class":509,"line":105},[507,87075,86878],{"class":572},[507,87077,86881],{"class":730},[507,87079,86884],{"class":517},[507,87081,87082,87084,87086],{"class":509,"line":540},[507,87083,1944],{"class":572},[507,87085,86891],{"class":730},[507,87087,86884],{"class":517},[507,87089,87090,87092,87094,87096],{"class":509,"line":553},[507,87091,86898],{"class":576},[507,87093,86901],{"class":730},[507,87095,86904],{"class":730},[507,87097,87098],{"class":730}," ionq-hardware-results.py\n",[13,87100,87102],{"id":87101},"celebrate","Celebrate",[18,87104,87105,87106,53],{},"Have you received your hardware results yet? Yes? Well then you’ve accomplished a lot here: You’ve just harnessed the power of our quantum universe to compute a thousand entangled coin flips. Take a beat, savor the moment. More Qollab tutorials are on on the way. In the meantime check out IBM’s incredibly informative ",[49,87107,87109],{"href":87108},"https:\u002F\u002Fwww.youtube.com\u002F@qiskit","Qiskit YouTube channel",[953,87111,87112],{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":87114},[87115,87116,87117,87118,87119,87122,87125,87126],{"id":86424,"depth":105,"text":86425},{"id":86489,"depth":105,"text":86490},{"id":86528,"depth":105,"text":86529},{"id":86539,"depth":105,"text":86540},{"id":86811,"depth":105,"text":86812,"children":87120},[87121],{"id":86826,"depth":540,"text":86827},{"id":86848,"depth":105,"text":86849,"children":87123},[87124],{"id":86916,"depth":540,"text":86917},{"id":86930,"depth":105,"text":86931},{"id":87101,"depth":105,"text":87102},[112,85993,85994,86361],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Stewart Smith","Creative technologist","Run a quantum circuit on IonQ’s quantum hardware and receive the results asynchronously. Builds upon our previous tutorials for executing quantum circuits on IonQ’s cloud simulator, and installing Python via UV.","Run a quantum circuit on IonQ's real hardware and get the results asynchronously, building on our earlier lessons on the IonQ cloud simulator.",{"image":87135,"alt":86361},"\u002F_content\u002Fimages\u002Fionq-hardware\u002Fhero.webp",{},{"slug":86352,"title":87138,"desc":87139},"6 · Create a Project on Qollab","This guide highlights the steps to build and publish your project; a step-by-step walkthrough of Qollab's project creation process, from naming…","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-hardware","2026-01-09",[],{"title":87144,"description":87133},"Run circuits on IonQ hardware · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-hardware",[],"GvDMPLz_bZ84G9vBG12vZ-23EjSejtbi3kt3tYwzdoQ",{"id":87149,"title":87150,"authors":87151,"body":87152,"breadcrumb":88401,"builders":88402,"byline":88403,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":88404,"description":88405,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":88406,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":88408,"navigation":133,"newsItems":116,"next":88409,"ogImage":116,"order":553,"outcomes":116,"path":88412,"publishDate":88413,"readingTime":88414,"related":88415,"relatedProjects":116,"seo":88416,"stem":88418,"tags":88419,"track":86016,"trackName":85994,"__hash__":88420},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup.md","Setup and simulate with IonQ",[86363],{"type":10,"value":87153,"toc":88380},[87154,87193,87197,87208,87211,87215,87219,87230,87234,87237,87246,87250,87254,87274,87285,87324,87336,87340,87343,87361,87364,87383,87387,87411,87415,87418,87436,87439,87458,87462,87468,87481,87483,87498,87501,87505,87511,87529,87535,87539,87546,87560,87563,87597,87600,87671,87675,87699,87711,87719,87723,87733,87740,87744,87749,87806,87817,87826,87829,87850,87856,87873,87881,87885,87901,88051,88055,88058,88122,88133,88136,88170,88173,88228,88231,88256,88260,88270,88352,88356,88377],[18,87155,87156,86371,87158,86376,87160,86381,87162,622,87166,622,87169,622,87173,87177,87178,87182,87183,87187,87188,87190,87191,53],{},[49,87157,76863],{"href":86370},[49,87159,86375],{"href":86374},[49,87161,86380],{"href":86379},[49,87163,87165],{"href":87164},"#_1-create-an-ionq-account","create an IonQ account",[49,87167,86516],{"href":87168},"#_2-generate-an-ionq-api-key",[49,87170,87172],{"href":87171},"#_3-create-a-new-project","create a new Python project",[49,87174,87176],{"href":87175},"#install-ionqs-sdk","install IonQ’s SDK"," (which is built upon ",[49,87179,87181],{"href":87180},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fqiskit-setup","IBM’s Qiskit","), and run an example quantum circuit on ",[49,87184,87186],{"href":87185},"#_6-run-your-circuit-on-ionqs-cloud-simulator","IonQ’s cloud simulator",". In order to follow along, it is essential that you have completed our previous guide for ",[49,87189,86411],{"href":86410},", which also covers operating a shell command-line interface. For additional information on IonQ’s SDK, see ",[49,87192,86421],{"href":86420},[13,87194,87196],{"id":87195},"_1-create-an-ionq-account","1. Create an IonQ account",[18,87198,87199,87200,87203,87204,87207],{},"Together, we are going to code a simple quantum circuit and execute it on IonQ’s cloud-based simulator. In order to connect to IonQ’s servers we must have authorization, and that comes in the form of an IonQ API key. To generate an API key, we must have an IonQ user account. Sign up for a new IonQ account by visiting ",[49,87201,87202],{"href":87202},"https:\u002F\u002Fcloud.ionq.com",". Click on the ",[154,87205,87206],{},"“Get started for free”"," link. From there you can create your account using an email address and password combination.",[86481,87209],{"href":87202,"label":87210},"Sign up for IonQ →",[831,87212],{"alt":104,"caption":87213,"no":104,"src":87214},"Screen grab of the IonQ signup page with the “Get started for free” link circled in orange by hand.","\u002F_content\u002Fimages\u002Fionq-setup\u002Fionq-signin.webp",[13,87216,87218],{"id":87217},"_2-generate-an-ionq-api-key","2. Generate an IonQ API key",[18,87220,87221,87222,87225,87226,87229],{},"Now that you have created an IonQ account, visit the “API Keys” tab of your account settings page: ",[49,87223,87224],{"href":87224},"https:\u002F\u002Fcloud.ionq.com\u002Fsettings\u002Fkeys",". Log in if you are not presently signed in, then click on the “",[154,87227,87228],{},"Generate key","” button.",[831,87231],{"alt":104,"caption":87232,"no":104,"src":87233},"Screen grab of IonQ’s “API Keys”tab of the account settings page with the “Generate key” button in the upper-right corner of the interface, circled in orange by hand and a hand-drawn orange arrow pointing to it.","\u002F_content\u002Fimages\u002Fionq-setup\u002Fionq-generate-button.webp",[18,87235,87236],{},"You will be prompted to provide a descriptive name for your key, and to choose an associated project. Note that using special characters in the description field may prevent the “Generate key” button from enabling itself. If you have not created any projects within your IonQ workspace, or have not been added to another account’s IonQ workspace project, you can always choose “Personal Workspace” as your key’s associated project.",[87238,87239,87240],"blockquote",{},[18,87241,87242,87245],{},[154,87243,87244],{},"Caution",": Your API key will only be revealed to you this once. Store it somewhere private and treat it as you would a regular password.",[831,87247],{"alt":104,"caption":87248,"no":104,"src":87249},"Screen grab of the generated key modal dialogue box (with the actual API key text obscured here).","\u002F_content\u002Fimages\u002Fionq-setup\u002Fionq-key-generated.webp",[13,87251,87253],{"id":87252},"_3-create-a-new-project","3. Create a new project",[18,87255,87256,87257,87259,87260,87263,87264,87268,87269,87273],{},"We’re going to create a new Python project that resides locally on our own machine and communicates with IonQ’s servers. If you have not already followed our guide to ",[49,87258,86411],{"href":86410},", do that now, and then return to this step. That guide explains the easy installation process, and also provides some familiarity with entering basic commands into a shell command-line interface. (If you are currently a ",[504,87261,87262],{},"venv"," user, ",[49,87265,87267],{"href":87266},"https:\u002F\u002Fconda.org\u002F","Conda"," user, or are accustomed to using unmanaged Python, we still strongly encourage you to switch to ",[49,87270,87272],{"href":87271},"https:\u002F\u002Fdocs.astral.sh\u002Fuv\u002F","UV",". Your future self will thank you.)",[18,87275,87276,87277,87280,87281,87284],{},"Create a new folder on your Desktop titled ",[504,87278,87279],{},"qollab-ionq",". (The exact name and location of this folder doesn’t matter so much, as long as it’s easy for you to access and work with.) Open a new shell prompt. ",[154,87282,87283],{},"Be sure to navigate to inside your project’s folder",", then enter the following command:",[498,87286,87288],{"className":86859,"code":87287,"language":86861,"meta":104,"style":104},"uv python install 3.12; uv python pin 3.12; uv init --app\n",[504,87289,87290],{"__ignoreMap":104},[507,87291,87292,87294,87296,87299,87302,87305,87307,87309,87312,87314,87316,87318,87321],{"class":509,"line":510},[507,87293,86898],{"class":576},[507,87295,86904],{"class":730},[507,87297,87298],{"class":730}," install",[507,87300,87301],{"class":583}," 3.12",[507,87303,87304],{"class":517},"; ",[507,87306,86898],{"class":576},[507,87308,86904],{"class":730},[507,87310,87311],{"class":730}," pin",[507,87313,87301],{"class":583},[507,87315,87304],{"class":517},[507,87317,86898],{"class":576},[507,87319,87320],{"class":730}," init",[507,87322,87323],{"class":583}," --app\n",[18,87325,87326,87327,87329,87330,87333,87334,53],{},"This will ensure that a Qiskit-compatible version of Python is installed and that our project is “pinned” to this version. (As of this writing, January 2026, Python 3.12 is the latest release that is fully compatible with the Qiskit SDK core, its various add-on packages that we will use in upcoming tutorials, and IonQ’s SDK. If you are one of those folks that becomes itchy at the prospect of not using the absolute lastest version of Python, be our guest. ",[49,87328,87272],{"href":87271}," makes it quick and easy to switch Python versions.) Finally, the ",[504,87331,87332],{},"init"," command initializes our app, creating several useful default files. If you experience trouble with this step, refer to our more detailed guide to ",[49,87335,86411],{"href":86410},[2513,87337,87339],{"id":87338},"install-ionqs-sdk","Install IonQ’s SDK",[18,87341,87342],{},"IonQ’s SDK is built upon IBM’s Qiskit, allowing us to take advantage of Qiskit’s flourishing ecosystem and IonQ’s unqiue quantum hardware. Enter the following command into our shell to install the latest versions of both IBM’s Qiskit SDK and IonQ’s SDK:",[498,87344,87346],{"className":86859,"code":87345,"language":86861,"meta":104,"style":104},"uv add qiskit qiskit-ionq\n",[504,87347,87348],{"__ignoreMap":104},[507,87349,87350,87352,87355,87358],{"class":509,"line":510},[507,87351,86898],{"class":576},[507,87353,87354],{"class":730}," add",[507,87356,87357],{"class":730}," qiskit",[507,87359,87360],{"class":730}," qiskit-ionq\n",[18,87362,87363],{},"Once this process is complete you can enter the following command into our shell to perform an optional sanity check. If all’s gone well, our shell will respond with an IonQ SDK version number.",[498,87365,87367],{"className":86859,"code":87366,"language":86861,"meta":104,"style":104},"uv run python -c \"import qiskit_ionq; print('IonQ SDK', qiskit_ionq.__version__)\"\n",[504,87368,87369],{"__ignoreMap":104},[507,87370,87371,87373,87375,87377,87380],{"class":509,"line":510},[507,87372,86898],{"class":576},[507,87374,86901],{"class":730},[507,87376,86904],{"class":730},[507,87378,87379],{"class":583}," -c",[507,87381,87382],{"class":730}," \"import qiskit_ionq; print('IonQ SDK', qiskit_ionq.__version__)\"\n",[13,87384,87386],{"id":87385},"_4-handle-your-ionq-api-key","4. Handle your IonQ API key",[18,87388,87389,87390,87393,87394,87397,87398,87401,87402,87405,87406,87410],{},"In order to communicate with IonQ’s servers, our new Python project requires your ",[49,87391,87392],{"href":87168},"IonQ API key",". (Recall that your API key is akin to a password, and should be treated as such.) IonQ’s ",[504,87395,87396],{},"IonQProvider"," package will ",[1031,87399,87400],{},"automatically"," look for an environment variable named ",[504,87403,87404],{},"IONQ_API_KEY",", and we have a few options for safely providing this. (See also ",[49,87407,87409],{"href":87408},"https:\u002F\u002Fdocs.ionq.com\u002Fguides\u002Fmanaging-api-keys","IonQ’s own guide to managing API keys",".)",[2513,87412,87414],{"id":87413},"set-a-key-for-this-shell-session","Set a key for this shell session",[18,87416,87417],{},"This is a quick, temporary solution that will make your API key available to our current shell session. (That means if we close our current shell and open a new one, or execute our Python script from within a different shell session than the one we’ve set your key in, your key value won’t be available to the Python script.) On macOS or Linux, enter the following into our shell:",[498,87419,87421],{"className":86859,"code":87420,"language":86861,"meta":104,"style":104},"export IONQ_API_KEY=\"your_real_key_here\"\n",[504,87422,87423],{"__ignoreMap":104},[507,87424,87425,87428,87431,87433],{"class":509,"line":510},[507,87426,87427],{"class":513},"export",[507,87429,87430],{"class":58306}," IONQ_API_KEY",[507,87432,573],{"class":572},[507,87434,87435],{"class":730},"\"your_real_key_here\"\n",[18,87437,87438],{},"Or on Windows, enter the following into PowerShell:",[498,87440,87442],{"className":86859,"code":87441,"language":86861,"meta":104,"style":104},"$Env:IONQ_API_KEY==\"your_real_key_here\"\n",[504,87443,87444],{"__ignoreMap":104},[507,87445,87446,87449,87451,87453,87455],{"class":509,"line":510},[507,87447,87448],{"class":58306},"$Env",[507,87450,24985],{"class":517},[507,87452,87404],{"class":58306},[507,87454,573],{"class":572},[507,87456,87457],{"class":730},"=\"your_real_key_here\"\n",[2513,87459,87461],{"id":87460},"confirm-your-key-is-present","Confirm your key is present",[18,87463,87464,87465,87467],{},"Regardless of whether we set your key only for this current shell session, or for every shell session that your user profile initiates, your key must be present in order to be read by IonQ’s ",[504,87466,87396],{}," package. We can confirm its presence on macOS or Linux by entering the following into our shell:",[498,87469,87471],{"className":86859,"code":87470,"language":86861,"meta":104,"style":104},"echo $IONQ_API_KEY\n",[504,87472,87473],{"__ignoreMap":104},[507,87474,87475,87478],{"class":509,"line":510},[507,87476,87477],{"class":572},"echo",[507,87479,87480],{"class":58306}," $IONQ_API_KEY\n",[18,87482,87438],{},[498,87484,87486],{"className":86859,"code":87485,"language":86861,"meta":104,"style":104},"echo $Env:IONQ_API_KEY\n",[504,87487,87488],{"__ignoreMap":104},[507,87489,87490,87492,87495],{"class":509,"line":510},[507,87491,87477],{"class":572},[507,87493,87494],{"class":58306}," $Env",[507,87496,87497],{"class":730},":IONQ_API_KEY\n",[18,87499,87500],{},"Our shell should respond with your IonQ API key.",[2513,87502,87504],{"id":87503},"confirm-that-your-key-is-functional","Confirm that your key is functional",[18,87506,87507,87508,87510],{},"Just because ",[504,87509,87404],{}," is available in our environment and contains a value doesn’t necessarily mean we’re authorized to access IonQ’s servers. Enter the following into our shell to confirm that we have access to various IonQ backends:",[498,87512,87514],{"className":86859,"code":87513,"language":86861,"meta":104,"style":104},"uv run python -c \"from qiskit_ionq import IonQProvider; p=IonQProvider(); print([b.name for b in p.backends()])\"\n",[504,87515,87516],{"__ignoreMap":104},[507,87517,87518,87520,87522,87524,87526],{"class":509,"line":510},[507,87519,86898],{"class":576},[507,87521,86901],{"class":730},[507,87523,86904],{"class":730},[507,87525,87379],{"class":583},[507,87527,87528],{"class":730}," \"from qiskit_ionq import IonQProvider; p=IonQProvider(); print([b.name for b in p.backends()])\"\n",[18,87530,87531,87532,87534],{},"Our shell should respond with a list of backends available to your account. Regardless of whether your account is on a free tier or paid tier, you should see a ",[1031,87533,21959],{}," profile in this list. If your account in on a paid tier you might also see available hardware profiles. (A free tier account may see a single generic hardware profile that serves as a placeholder, but you will be unable to send jobs to any actual quantum hardware.) If your key is missing or invalid then you will receive an authorization error rather than a list of backend profiles.",[2513,87536,87538],{"id":87537},"include-your-key-in-our-project","Include your key in our project",[18,87540,87541,87542,87545],{},"We’ve entered your key into our shell’s environment and confirmed that it functions. But what about the next time we open a new shell window? Wouldn’t it be easier if going forward our project always had access to your API key? We can accomplish this by creating a hidden “environment file” for our project that will seed our shell environment with your API key (and whatever other variables we may wish to set). Create a ",[504,87543,87544],{},".env"," file inside of our project’s folder, add the following line, and save the file:",[498,87547,87548],{"className":86859,"code":87420,"language":86861,"meta":104,"style":104},[504,87549,87550],{"__ignoreMap":104},[507,87551,87552,87554,87556,87558],{"class":509,"line":510},[507,87553,87427],{"class":513},[507,87555,87430],{"class":58306},[507,87557,573],{"class":572},[507,87559,87435],{"class":730},[18,87561,87562],{},"On macOS or Linux, when we’re ready to use this key we would enter the following into our shell to load our environment file and run a Python script.",[498,87564,87566],{"className":86859,"code":87565,"language":86861,"meta":104,"style":104},"set -a; source .\u002F.env; set +a; uv run python our-future-script.py\n",[504,87567,87568],{"__ignoreMap":104},[507,87569,87570,87572,87574,87576,87578,87580,87582,87584,87586,87588,87590,87592,87594],{"class":509,"line":510},[507,87571,1944],{"class":572},[507,87573,86870],{"class":583},[507,87575,87304],{"class":517},[507,87577,86878],{"class":572},[507,87579,86881],{"class":730},[507,87581,87304],{"class":517},[507,87583,1944],{"class":572},[507,87585,86891],{"class":730},[507,87587,87304],{"class":517},[507,87589,86898],{"class":576},[507,87591,86901],{"class":730},[507,87593,86904],{"class":730},[507,87595,87596],{"class":730}," our-future-script.py\n",[18,87598,87599],{},"Or in Windows PowerShell we would do the following:",[498,87601,87603],{"className":86859,"code":87602,"language":86861,"meta":104,"style":104},"Get-Content .env | ForEach-Object { if ($_ -match '^\\s*([^#=]+?)\\s*=\\s*(.*)\\s*$') { Set-Item -Path \"Env:$($matches[1])\" -Value $matches[2] }}\nuv run python our-future-script.py\n",[504,87604,87605,87661],{"__ignoreMap":104},[507,87606,87607,87610,87613,87616,87619,87621,87623,87626,87629,87632,87635,87638,87641,87644,87647,87650,87653,87656,87658],{"class":509,"line":510},[507,87608,87609],{"class":576},"Get-Content",[507,87611,87612],{"class":730}," .env",[507,87614,87615],{"class":517}," | ",[507,87617,87618],{"class":576},"ForEach-Object",[507,87620,73290],{"class":730},[507,87622,66162],{"class":730},[507,87624,87625],{"class":517}," ($_ ",[507,87627,87628],{"class":583},"-match",[507,87630,87631],{"class":730}," '^\\s*([^#=]+?)\\s*=\\s*(.*)\\s*$'",[507,87633,87634],{"class":517},") { ",[507,87636,87637],{"class":576},"Set-Item",[507,87639,87640],{"class":583}," -Path",[507,87642,87643],{"class":730}," \"Env:$(",[507,87645,87646],{"class":58306},"$matches",[507,87648,87649],{"class":730},"[1])\"",[507,87651,87652],{"class":583}," -Value",[507,87654,87655],{"class":58306}," $matches",[507,87657,8547],{"class":730},[507,87659,87660],{"class":730}," }}\n",[507,87662,87663,87665,87667,87669],{"class":509,"line":105},[507,87664,86898],{"class":576},[507,87666,86901],{"class":730},[507,87668,86904],{"class":730},[507,87670,87596],{"class":730},[7660,87672,87674],{"id":87673},"protect-your-api-key","Protect your API key",[18,87676,87677,87678,87680,87681,87684,87685,87687,87688,87690,87691,87694,87695,87698],{},"Perhaps our project is part of a shared code repository. The last thing we want is to accidentally publish your private API key along with the codebase. In that case, add ",[504,87679,87544],{}," to our ",[504,87682,87683],{},".gitignore"," list. (If the ",[504,87686,87683],{}," does not exist within our project folder, create it, enter the text ",[504,87689,87544],{}," on a single line, and save it.) As a courtesy to our teammates and future selves, also create an ",[1031,87692,87693],{},"example"," environment file that serves as a reminder and template for what information must be provided in order for the project to function. Name this file ",[504,87696,87697],{},".env.example"," and include the following line in it:",[498,87700,87702],{"className":86859,"code":87701,"language":86861,"meta":104,"style":104},"IONQ_API_KEY=\n",[504,87703,87704],{"__ignoreMap":104},[507,87705,87706,87708],{"class":509,"line":510},[507,87707,87404],{"class":58306},[507,87709,87710],{"class":572},"=\n",[18,87712,87713,87714,56374,87716,87718],{},"Now our teammates (or future us) can simply copy this file from ",[504,87715,87697],{},[504,87717,87544],{}," and fill in the appropriate information locally. (This is a standard “don’t leak tokens” pattern.)",[13,87720,87722],{"id":87721},"_5-update-our-transpile-settings","5. Update our transpile settings",[18,87724,87725,87726,87729,87730],{},"There’s always a gap between the theoretical and the actual, between the clear expression of intent and the dirty business of actually making something ",[1031,87727,87728],{},"function."," When we code a quantum circuit, more often than not we are creating idealistically. Part of Qiskit’s magic is that it transforms our quantum circuit design in two ways: It reconfigures our gates for IBM’s specific quantum hardware architecture, and also optimizes our algorithms for maximum efficiency, also based on IBM’s available hardware. This optimization process can reduce register depth, merge gates, and change the overall circuit structure while preserving its semantics. The process of compiling code for one model, then translating that compilation to function on a different model, is called ",[1031,87731,87732],{},"transpiling.",[18,87734,87735,87736,87739],{},"The twist with our scenario is that we are ",[1031,87737,87738],{},"not using IBM’s architecture"," to execute our quantum circuit. We are instead using’s IonQ’s architecture. IonQ’s SDK has its own methods for translating and optimizing our circuit designs, based on its own unique quantum hardware. We want IonQ to have direct access to our original circuit design, not a version that has been “optimized” for some other architecture. (That would be like working from a lossy copy when we have access to the original right in front of us.) In order to hand IonQ’s SDK our original, unaltered circuit design, we must tell Qiskit to only make minimal, necessary changes.",[2513,87741,87743],{"id":87742},"qiskits-transpile-levels","Qiskit’s transpile levels",[18,87745,87746,87747,53],{},"The following table describes each of Qiskit’s transpile levels, its general approach to optimization, and what it’s best suited for. We’re most interested in creating “early experiments” and “transpiling at scale”, and will opt for an optimization level of ",[504,87748,625],{},[41852,87750,87751,87765],{},[41855,87752,87753],{},[41858,87754,87755,87759,87762],{},[41861,87756,87758],{"align":87757},"center","Level",[41861,87760,87761],{},"Optimization",[41861,87763,87764],{},"Best suited for",[41868,87766,87767,87776,87786,87796],{},[41858,87768,87769,87771,87773],{},[41873,87770,601],{"align":87757},[41873,87772,56764],{},[41873,87774,87775],{},"Hardware-native backends (IonQ, neutral atoms), learning, debugging, preserving algorithm structure.",[41858,87777,87778,87780,87783],{},[41873,87779,625],{"align":87757},[41873,87781,87782],{},"Light",[41873,87784,87785],{},"Early experiments, hardware with mild noise, transpiling at scale.",[41858,87787,87788,87790,87793],{},[41873,87789,584],{"align":87757},[41873,87791,87792],{},"Medium",[41873,87794,87795],{},"General IBM hardware usage, balanced workflows.",[41858,87797,87798,87800,87803],{},[41873,87799,8226],{"align":87757},[41873,87801,87802],{},"Heavy",[41873,87804,87805],{},"Final production runs on noisy IBM devices.",[18,87807,87808,87809,87812,87813,87816],{},"To ensure that all of our future IonQ circuits run as expected, we can edit Qiskit’s user configuration file. The Qiskit installation process creates a hidden ",[504,87810,87811],{},".qiskit"," folder within your home folder. We need to create (or edit) a ",[504,87814,87815],{},"settings.conf"," file within that folder. On macOS or Linux that location should be:",[498,87818,87820],{"className":86859,"code":87819,"language":86861,"meta":104,"style":104},"~\u002F.qiskit\u002Fsettings.conf\n",[504,87821,87822],{"__ignoreMap":104},[507,87823,87824],{"class":509,"line":510},[507,87825,87819],{"class":517},[18,87827,87828],{},"On Windows that location should be:",[498,87830,87832],{"className":86859,"code":87831,"language":86861,"meta":104,"style":104},"$HOME\\.qiskit\\settings.conf\n",[504,87833,87834],{"__ignoreMap":104},[507,87835,87836,87839,87842,87844,87847],{"class":509,"line":510},[507,87837,87838],{"class":58306},"$HOME",[507,87840,87841],{"class":572},"\\.",[507,87843,86107],{"class":517},[507,87845,87846],{"class":572},"\\s",[507,87848,87849],{"class":517},"ettings.conf\n",[18,87851,87852,87853,87855],{},"Open (or create) that ",[504,87854,87815],{}," file, add the following two lines of code, then save the file:",[498,87857,87861],{"className":87858,"code":87859,"language":87860,"meta":104,"style":104},"language-conf shiki shiki-themes one-dark-pro","[default]\ntranspile_optimization_level = 1\n","conf",[504,87862,87863,87868],{"__ignoreMap":104},[507,87864,87865],{"class":509,"line":510},[507,87866,87867],{},"[default]\n",[507,87869,87870],{"class":509,"line":105},[507,87871,87872],{},"transpile_optimization_level = 1\n",[18,87874,87875,87876,87880],{},"This will prevent Qiskit from making aggressive rewrites to our circuit design, handing it off cleanly to IonQ’s own transpiler. (It will also prevent a warning from IonQ’s SDK when we run our quantum circuit just a bit further down in this tutorial.) For additional information, see IonQ’s article “",[49,87877,87879],{"href":87878},"https:\u002F\u002Fdocs.ionq.com\u002Fsdks\u002Fqiskit\u002Fnative-gates-qiskit","Compilation and native gates with Qiskit",".”",[13,87882,87884],{"id":87883},"_6-run-your-circuit-on-ionqs-cloud-simulator","6. Run your circuit on IonQ’s cloud simulator",[18,87886,87887,87888,86508,87891,87893,87894,87897,87898,87900],{},"With our ",[49,87889,87890],{"href":86410},"UV-initiated project folder",[49,87892,87392],{"href":87168}," in place, we’re ready to use IonQ’s quantum ",[1031,87895,87896],{},"simulator."," Create a new blank file within your project folder named ",[504,87899,86550],{},", open it with your source code editor, paste the following code into it, and save the file:",[498,87902,87904],{"className":500,"code":87903,"language":502,"meta":104,"style":104},"from qiskit import QuantumCircuit\nfrom qiskit_ionq import IonQProvider\n\nqc = QuantumCircuit( 2, name=\"Bell state on IonQ simulator\" )\nqc.h( 0 )\nqc.cx( 0, 1 )\nqc.measure_all()\n\nprovider = IonQProvider()\nbackend  = provider.get_backend( \"simulator\" )\njob = backend.run( qc, shots=1000 )\nprint( job.get_counts() )\n",[504,87905,87906,87916,87926,87930,87953,87965,87981,87989,87993,88003,88020,88041],{"__ignoreMap":104},[507,87907,87908,87910,87912,87914],{"class":509,"line":510},[507,87909,529],{"class":513},[507,87911,532],{"class":517},[507,87913,514],{"class":513},[507,87915,537],{"class":517},[507,87917,87918,87920,87922,87924],{"class":509,"line":105},[507,87919,529],{"class":513},[507,87921,86577],{"class":517},[507,87923,514],{"class":513},[507,87925,86970],{"class":517},[507,87927,87928],{"class":509,"line":540},[507,87929,556],{"emptyLinePlaceholder":133},[507,87931,87932,87934,87936,87938,87940,87942,87944,87946,87948,87951],{"class":509,"line":553},[507,87933,569],{"class":517},[507,87935,573],{"class":572},[507,87937,577],{"class":576},[507,87939,86603],{"class":517},[507,87941,584],{"class":583},[507,87943,622],{"class":517},[507,87945,22008],{"class":2155},[507,87947,573],{"class":572},[507,87949,87950],{"class":730},"\"Bell state on IonQ simulator\"",[507,87952,86624],{"class":517},[507,87954,87955,87957,87959,87961,87963],{"class":509,"line":559},[507,87956,593],{"class":517},[507,87958,596],{"class":576},[507,87960,86603],{"class":517},[507,87962,601],{"class":583},[507,87964,86624],{"class":517},[507,87966,87967,87969,87971,87973,87975,87977,87979],{"class":509,"line":566},[507,87968,593],{"class":517},[507,87970,615],{"class":576},[507,87972,86603],{"class":517},[507,87974,601],{"class":583},[507,87976,622],{"class":517},[507,87978,625],{"class":583},[507,87980,86624],{"class":517},[507,87982,87983,87985,87987],{"class":509,"line":590},[507,87984,593],{"class":517},[507,87986,86659],{"class":576},[507,87988,781],{"class":517},[507,87990,87991],{"class":509,"line":610},[507,87992,556],{"emptyLinePlaceholder":133},[507,87994,87995,87997,87999,88001],{"class":509,"line":634},[507,87996,86670],{"class":517},[507,87998,573],{"class":572},[507,88000,86582],{"class":576},[507,88002,781],{"class":517},[507,88004,88005,88007,88009,88011,88013,88015,88018],{"class":509,"line":661},[507,88006,86989],{"class":517},[507,88008,573],{"class":572},[507,88010,86685],{"class":517},[507,88012,86688],{"class":576},[507,88014,86603],{"class":517},[507,88016,88017],{"class":730},"\"simulator\"",[507,88019,86624],{"class":517},[507,88021,88022,88024,88026,88028,88030,88033,88035,88037,88039],{"class":509,"line":678},[507,88023,23964],{"class":517},[507,88025,573],{"class":572},[507,88027,73487],{"class":517},[507,88029,22501],{"class":576},[507,88031,88032],{"class":517},"( qc, ",[507,88034,68762],{"class":2155},[507,88036,573],{"class":572},[507,88038,79870],{"class":583},[507,88040,86624],{"class":517},[507,88042,88043,88045,88047,88049],{"class":509,"line":683},[507,88044,8525],{"class":572},[507,88046,87041],{"class":517},[507,88048,73558],{"class":576},[507,88050,86791],{"class":517},[2513,88052,88054],{"id":88053},"our-first-ionq-bell-state","Our first IonQ Bell state",[18,88056,88057],{},"Before we run our short Python script above, let’s break down what it intends to accomplish. First, we import Qiskit’s tools for describing quantum circuits. Then we import IonQ’s interface for talking to various resources for executing quantum circuits. These two imports are all we need to begin our circuit building journey.",[18,88059,88060,88061,88064,88065,10799,88069,88073,88074,88077,88078,88081,88082,88086,88087,88089,88090,88094,88095,88081,88098,88102,88103,88105,88106,88108,88109,88113,88114,88116,88117,88121],{},"The command ",[504,88062,88063],{},"QuantumCircuit( 2, … )"," creates a quantum circuit composed of two qubit registers, both initialized to a | 0 ⟩ (“ket zero”) state. (Need a refresher on ",[49,88066,88068],{"href":88067},"\u002Flearn\u002Fquantum-foundations\u002Fqubits","qubits",[49,88070,88072],{"href":88071},"\u002Flearn\u002Fquantum-foundations\u002Fqubits#ket-notation","ket notation","? See our ",[49,88075,88076],{"href":88067},"guide to qubits",".) We use the command ",[504,88079,88080],{},"qc.h( 0 )"," to place a ",[49,88083,88085],{"href":88084},"\u002Flearn\u002Fquantum-foundations\u002Fgates#hadamard-gate","Hadamard gate"," onto register ",[504,88088,601],{},", flipping that qubit into ",[49,88091,88093],{"href":88092},"\u002Flearn\u002Fquantum-foundations\u002Fqubits#superposition","superposition",". Next, we use the command ",[504,88096,88097],{},"qc.cx( 0, 1 )",[49,88099,88101],{"href":88100},"\u002Flearn\u002Fquantum-foundations\u002Fgates#controlled-not-gate","CNOT gate"," across the two registers, using register ",[504,88104,601],{}," as the control qubit and register ",[504,88107,625],{}," as the target qubit. This creates a ",[49,88110,88112],{"href":88111},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBell_state","Bell state",", distributing register ",[504,88115,601],{},"’s superposition across the two qubits, ",[49,88118,88120],{"href":88119},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_entanglement","entangling"," them.",[18,88123,88124,88125,88128,88129,88132],{},"Finally, we use ",[504,88126,88127],{},"qc.measure_all()"," to measure both qubit registers, collapsing the distributed superposition into a definitive value. The two possible measured values are | 00 ⟩ and | 11 ⟩ , each with a 50% probabilty of occurrence. This simulation is run 1,000 times (",[504,88130,88131],{},"shots=1000",") and the results of these “shots” are output to our shell. While it is extremely unlikely that your result will divide into exactly 500 of each value, it should arrive reasonably close.",[18,88134,88135],{},"Now that we have a better idea of what our script does, let’s execute it. On macOS or Linux, enter the following into our shell to load our environment variables and execute our script:",[498,88137,88139],{"className":86859,"code":88138,"language":86861,"meta":104,"style":104},"set -a; source .\u002F.env; set +a; uv run python ionq-simulator.py\n",[504,88140,88141],{"__ignoreMap":104},[507,88142,88143,88145,88147,88149,88151,88153,88155,88157,88159,88161,88163,88165,88167],{"class":509,"line":510},[507,88144,1944],{"class":572},[507,88146,86870],{"class":583},[507,88148,87304],{"class":517},[507,88150,86878],{"class":572},[507,88152,86881],{"class":730},[507,88154,87304],{"class":517},[507,88156,1944],{"class":572},[507,88158,86891],{"class":730},[507,88160,87304],{"class":517},[507,88162,86898],{"class":576},[507,88164,86901],{"class":730},[507,88166,86904],{"class":730},[507,88168,88169],{"class":730}," ionq-simulator.py\n",[18,88171,88172],{},"Or in Windows PowerShell enter the following:",[498,88174,88176],{"className":86859,"code":88175,"language":86861,"meta":104,"style":104},"Get-Content .env | ForEach-Object { if ($_ -match '^\\s*([^#=]+?)\\s*=\\s*(.*)\\s*$') { Set-Item -Path \"Env:$($matches[1])\" -Value $matches[2] }}\nuv run python ionq-simulator.py\n",[504,88177,88178,88218],{"__ignoreMap":104},[507,88179,88180,88182,88184,88186,88188,88190,88192,88194,88196,88198,88200,88202,88204,88206,88208,88210,88212,88214,88216],{"class":509,"line":510},[507,88181,87609],{"class":576},[507,88183,87612],{"class":730},[507,88185,87615],{"class":517},[507,88187,87618],{"class":576},[507,88189,73290],{"class":730},[507,88191,66162],{"class":730},[507,88193,87625],{"class":517},[507,88195,87628],{"class":583},[507,88197,87631],{"class":730},[507,88199,87634],{"class":517},[507,88201,87637],{"class":576},[507,88203,87640],{"class":583},[507,88205,87643],{"class":730},[507,88207,87646],{"class":58306},[507,88209,87649],{"class":730},[507,88211,87652],{"class":583},[507,88213,87655],{"class":58306},[507,88215,8547],{"class":730},[507,88217,87660],{"class":730},[507,88219,88220,88222,88224,88226],{"class":509,"line":105},[507,88221,86898],{"class":576},[507,88223,86901],{"class":730},[507,88225,86904],{"class":730},[507,88227,88169],{"class":730},[18,88229,88230],{},"Take a patient breath as IonQ’s cloud simulator processes our circuit. In a few moments we should receive results similar to the following:",[498,88232,88234],{"className":86859,"code":88233,"language":86861,"meta":104,"style":104},"{'00': 492, '11': 508}\n",[504,88235,88236],{"__ignoreMap":104},[507,88237,88238,88240,88243,88245,88248,88251,88254],{"class":509,"line":510},[507,88239,2810],{"class":517},[507,88241,88242],{"class":576},"'00'",[507,88244,24985],{"class":572},[507,88246,88247],{"class":730}," 492,",[507,88249,88250],{"class":730}," '11':",[507,88252,88253],{"class":583}," 508",[507,88255,23875],{"class":730},[2513,88257,88259],{"id":88258},"optional-use-ionqs-noisy-simulator","Optional: Use IonQ’s “noisy simulator”",[18,88261,88262,88263,88265,88266,88269],{},"In addition to its basic simulator, IonQ provides “noise models” that produce results closer to that of actual quantum hardware. Insert the highlighted line of code below into your existing ",[504,88264,86550],{}," as indicated to use IonQ’s ",[504,88267,88268],{},"aria-1"," noise model in your simulations:",[498,88271,88273],{"className":500,"code":88272,"language":502,"meta":104,"style":104},"provider = IonQProvider()\nbackend  = provider.get_backend( \"simulator\" )\nbackend.set_options( noise_model=\"aria-1\" )\njob = backend.run( qc, shots=1000 )\nprint( job.get_counts() )\n",[504,88274,88275,88285,88301,88322,88342],{"__ignoreMap":104},[507,88276,88277,88279,88281,88283],{"class":509,"line":510},[507,88278,86670],{"class":517},[507,88280,573],{"class":572},[507,88282,86582],{"class":576},[507,88284,781],{"class":517},[507,88286,88287,88289,88291,88293,88295,88297,88299],{"class":509,"line":105},[507,88288,86989],{"class":517},[507,88290,573],{"class":572},[507,88292,86685],{"class":517},[507,88294,86688],{"class":576},[507,88296,86603],{"class":517},[507,88298,88017],{"class":730},[507,88300,86624],{"class":517},[507,88302,88303],{"class":509,"line":540},[86584,88304,88305,88307,88310,88312,88315,88317,88320],{},[507,88306,22398],{"class":517},[507,88308,88309],{"class":576},"set_options",[507,88311,86603],{"class":517},[507,88313,88314],{"class":2155},"noise_model",[507,88316,573],{"class":572},[507,88318,88319],{"class":730},"\"aria-1\"",[507,88321,1116],{"class":517},[507,88323,88324,88326,88328,88330,88332,88334,88336,88338,88340],{"class":509,"line":553},[507,88325,23964],{"class":517},[507,88327,573],{"class":572},[507,88329,73487],{"class":517},[507,88331,22501],{"class":576},[507,88333,88032],{"class":517},[507,88335,68762],{"class":2155},[507,88337,573],{"class":572},[507,88339,79870],{"class":583},[507,88341,86624],{"class":517},[507,88343,88344,88346,88348,88350],{"class":509,"line":559},[507,88345,8525],{"class":572},[507,88347,87041],{"class":517},[507,88349,73558],{"class":576},[507,88351,86791],{"class":517},[13,88353,88355],{"id":88354},"next-steps","Next steps",[18,88357,88358,88359,622,88361,622,88363,622,88366,88368,88369,88371,88372,88376],{},"We’ve ",[49,88360,86446],{"href":87164},[49,88362,86450],{"href":87168},[49,88364,88365],{"href":87171},"created a new Python project",[49,88367,86503],{"href":87175},", and run an example quantum circuit on ",[49,88370,87186],{"href":87185},". (That’s not a bad run for today!) Our next goal is to ",[49,88373,88375],{"href":88374},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-hardware","run quantum circuits on IonQ’s actual quantum hardware",". Keep in mind that this will require a paid tier account and some patience. We must create a support request for accessing IonQ’s QPUs, and wait for that request to be honored. Then once we do have access we must be mindful that QPU jobs are queued for execution and our circuits may need to wait several hours behind previously queued jobs before it’s our turn. When you’re ready to take your quantum journey to the next level join us on the hardware side!",[953,88378,88379],{},"html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}",{"title":104,"searchDepth":105,"depth":105,"links":88381},[88382,88383,88384,88387,88393,88396,88400],{"id":87195,"depth":105,"text":87196},{"id":87217,"depth":105,"text":87218},{"id":87252,"depth":105,"text":87253,"children":88385},[88386],{"id":87338,"depth":540,"text":87339},{"id":87385,"depth":105,"text":87386,"children":88388},[88389,88390,88391,88392],{"id":87413,"depth":540,"text":87414},{"id":87460,"depth":540,"text":87461},{"id":87503,"depth":540,"text":87504},{"id":87537,"depth":540,"text":87538},{"id":87721,"depth":105,"text":87722,"children":88394},[88395],{"id":87742,"depth":540,"text":87743},{"id":87883,"depth":105,"text":87884,"children":88397},[88398,88399],{"id":88053,"depth":540,"text":88054},{"id":88258,"depth":540,"text":88259},{"id":88354,"depth":105,"text":88355},[112,85993,85994,87150],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Create and leverage a free IonQ account to run an example quantum circuit on IonQ’s cloud simulator. Builds upon our previous tutorials for installing Python via UV, and installing IBM’s Qiskit SDK.","Create a free IonQ account and run an example circuit on IonQ's cloud simulator, building on our earlier Python and Qiskit setup lessons.",{"image":88407,"alt":87150},"\u002F_content\u002Fimages\u002Fionq-setup\u002Fhero.webp",{},{"slug":87140,"title":88410,"desc":88411},"5 · Run circuits on IonQ’s quantum hardware","Run a quantum circuit on IonQ’s quantum hardware and receive the results asynchronously. Builds upon our previous tutorials for executing quantum…","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup","2026-01-02","18 min read",[],{"title":88417,"description":88405},"Setup and simulate with IonQ · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fionq-setup",[],"4CoeHtmKoj9tpDycD0ayHhQ5rCyf7l95VA1ixVRVub0",{"id":88422,"title":85994,"authors":88423,"body":88424,"breadcrumb":88431,"builders":88432,"byline":88433,"challenge":116,"courseAuthor":88434,"courseLead":88439,"dek":88440,"description":88441,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":88442,"lessonCount":116,"meta":88443,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":88444,"path":88449,"publishDate":88450,"readingTime":116,"related":88451,"relatedProjects":116,"seo":88452,"stem":88454,"tags":88455,"track":86016,"trackName":116,"__hash__":88456},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project.md",[86363],{"type":10,"value":88425,"toc":88429},[88426],[18,88427,88428],{},"This guide will walk you through downloading, installing, and enjoying Python via the UV package installer and project manager.",{"title":104,"searchDepth":105,"depth":105,"links":88430},[],[112,85993,85994],[],{"username":86363,"name":87130,"role":104,"avatar":104},{"name":87130,"role":104,"bio":88435,"avatar":104,"links":88436},"Stewart Smith is an award-winning creative technologist, artist, writer, and speaker. He’s led innovation teams across ten time zones, and wields wisdom accrued from his years with Google, Amazon, and Unity. Educated as a graphic designer (MFA, Yale University), his work has spanned quantum computing, artificial intelligence, spatial computing, aerospace, and fine art.",[88437],{"label":73068,"href":88438},"\u002Fu\u002Fstewart-smith","This course takes you from an empty machine to a published project. You set up Python, UV, and Qiskit locally, learn the Python you need, then get IonQ access and run your first circuit on real quantum hardware, and finally create a project on Qollab and use the Code Playground to share what you built.","Set up Python and Qiskit, run your first circuit on real IonQ hardware, then publish your project on Qollab.","A free seven-lesson course from local Python and Qiskit setup to running circuits on real IonQ hardware, and finally publishing your first project.","landing",{},[88445,88446,88447,88448],"How to set up a local Python, UV, and Qiskit toolchain","The Python basics that quantum programs are written with","How to simulate circuits with IonQ and then run them on real QPUs","How to create a project on Qollab and share it in the Code Playground","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project","2026-01-01",[],{"title":88453,"description":88441},"Building your first Qollab project · Learn by building","blog\u002Flearn\u002Fbuilding-your-first-qollab-project",[],"shemK7IIkLNkwz7pzkFXs38UbVn0muzXOHpQiSJGOfg",{"id":88458,"title":88459,"authors":88460,"body":88461,"breadcrumb":90095,"builders":90096,"byline":90097,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":90098,"description":90099,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":90100,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":90102,"navigation":133,"newsItems":116,"next":90103,"ogImage":116,"order":105,"outcomes":116,"path":90107,"publishDate":88450,"readingTime":90108,"related":90109,"relatedProjects":116,"seo":90110,"stem":90112,"tags":90113,"track":86016,"trackName":85994,"__hash__":90114},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-basics.md","Learn Python language basics",[86363],{"type":10,"value":88462,"toc":90065},[88463,88483,88487,88509,88527,88533,88547,88557,88567,88583,88587,88602,88645,88648,88654,88686,88709,88713,88746,88749,88846,88850,88989,88993,88997,89023,89027,89052,89056,89090,89093,89119,89122,89156,89159,89192,89196,89231,89237,89241,89315,89319,89356,89361,89365,89368,89372,89375,89420,89433,89481,89485,89532,89535,89592,89595,89655,89659,89662,89666,89687,89691,89714,89718,89743,89747,89771,89775,89810,89814,89843,89847,89850,89922,89926,89951,89954,89958,90062],[18,88464,88465,88466,88469,88470,88473,88474,88478,88479,88482],{},"This guide assumes that you have already installed the ",[49,88467,88468],{"href":87271},"UV package manager"," and latest version of ",[49,88471,496],{"href":88472},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPython%3C\u002Fem%3E(programming_language)"," on your machine. It also assumes that you are able to open a ",[49,88475,88477],{"href":88476},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FShell%3C\u002Fem%3E(computing)","shell application",", and are comfortable editing code. If any of that sounds unfamiliar or outside your expertise, don’t fret! We have a simple guide for that: ",[49,88480,88481],{"href":86410},"Setup Python on your machine",". Give that guide a thorough look over first, then come back here to start flexing your code muscles.",[13,88484,88486],{"id":88485},"running-python","Running Python",[18,88488,88489,88490,88494,88495,88498,88499,88501,88502,88505,88506,88508],{},"Python runs line-by-line, like a ",[49,88491,88493],{"href":88492},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRead%E2%80%93eval%E2%80%93print_loop","Read-Evaluate-Print Loop (REPL)",". Python scripts are just plain text files, often with file names ending in a ",[504,88496,88497],{},".py"," suffix. Our code examples rely on the ",[49,88500,88468],{"href":87271},". (See our guide for ",[49,88503,88504],{"href":86410},"Setting up Python and UV on your machine",".) A simple one-line Python script can be run from the command-line shell via ",[504,88507,86898],{}," like so:",[498,88510,88512],{"className":86859,"code":88511,"language":86861,"meta":104,"style":104},"uv run python -c \"print('Hello, World!')\"\n",[504,88513,88514],{"__ignoreMap":104},[507,88515,88516,88518,88520,88522,88524],{"class":509,"line":510},[507,88517,86898],{"class":576},[507,88519,86901],{"class":730},[507,88521,86904],{"class":730},[507,88523,87379],{"class":583},[507,88525,88526],{"class":730}," \"print('Hello, World!')\"\n",[18,88528,88529,88530,88532],{},"To enter a full Python REPL via ",[504,88531,86898],{},", enter the following into your shell:",[498,88534,88536],{"className":86859,"code":88535,"language":86861,"meta":104,"style":104},"uv run python\n",[504,88537,88538],{"__ignoreMap":104},[507,88539,88540,88542,88544],{"class":509,"line":510},[507,88541,86898],{"class":576},[507,88543,86901],{"class":730},[507,88545,88546],{"class":730}," python\n",[18,88548,88549,88550,947,88553,88556],{},"To exit the Python REPL, type ",[504,88551,88552],{},"exit()",[504,88554,88555],{},"quit()"," and press Enter. For Python 3.13 and later, those parenthesis may be omitted.",[18,88558,88559,88560,88562,88563,88566],{},"To run a Python script file via ",[504,88561,86898],{},", enter the following, replacing ",[504,88564,88565],{},"main.py"," with the name of the script you wish to execute.",[498,88568,88570],{"className":86859,"code":88569,"language":86861,"meta":104,"style":104},"uv run python main.py\n",[504,88571,88572],{"__ignoreMap":104},[507,88573,88574,88576,88578,88580],{"class":509,"line":510},[507,88575,86898],{"class":576},[507,88577,86901],{"class":730},[507,88579,86904],{"class":730},[507,88581,88582],{"class":730}," main.py\n",[13,88584,88586],{"id":88585},"basic-syntax","Basic Syntax",[18,88588,88589,88592,88593,622,88595,86110,88598,88601],{},[154,88590,88591],{},"Variable creation is implicit",". Unlike other scripting languages, there’s no ",[504,88594,86696],{},[504,88596,88597],{},"let",[504,88599,88600],{},"const"," keywords required in order to create a variable handle. Assignment binds a name to a value:",[498,88603,88605],{"className":500,"code":88604,"language":502,"meta":104,"style":104},"x = 10\ny = \"hello\"\nz = [1, 2, 3]\n",[504,88606,88607,88616,88625],{"__ignoreMap":104},[507,88608,88609,88611,88613],{"class":509,"line":510},[507,88610,69745],{"class":517},[507,88612,573],{"class":572},[507,88614,88615],{"class":583}," 10\n",[507,88617,88618,88620,88622],{"class":509,"line":105},[507,88619,69764],{"class":517},[507,88621,573],{"class":572},[507,88623,88624],{"class":730}," \"hello\"\n",[507,88626,88627,88629,88631,88633,88635,88637,88639,88641,88643],{"class":509,"line":540},[507,88628,71105],{"class":517},[507,88630,573],{"class":572},[507,88632,8427],{"class":517},[507,88634,625],{"class":583},[507,88636,622],{"class":517},[507,88638,584],{"class":583},[507,88640,622],{"class":517},[507,88642,8226],{"class":583},[507,88644,1794],{"class":517},[18,88646,88647],{},"Everything is an object. Variables are labels, not boxes.",[18,88649,88650,88653],{},[154,88651,88652],{},"Indentation"," (not curly braces or other visible characters) defines scope blocks:",[498,88655,88657],{"className":500,"code":88656,"language":502,"meta":104,"style":104},"if cond:\n    do_something()\nelse:\n    do_other()\n",[504,88658,88659,88666,88673,88679],{"__ignoreMap":104},[507,88660,88661,88663],{"class":509,"line":510},[507,88662,1645],{"class":513},[507,88664,88665],{"class":517}," cond:\n",[507,88667,88668,88671],{"class":509,"line":105},[507,88669,88670],{"class":576},"    do_something",[507,88672,781],{"class":517},[507,88674,88675,88677],{"class":509,"line":540},[507,88676,61407],{"class":513},[507,88678,1728],{"class":517},[507,88680,88681,88684],{"class":509,"line":553},[507,88682,88683],{"class":576},"    do_other",[507,88685,781],{"class":517},[18,88687,88688,88691,88692,88696,88697,88700,88701,88705,88706,88708],{},[154,88689,88690],{},"Indentation is semantic",", that is, your ",[49,88693,88695],{"href":88694},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNegative_space","negative space"," carries logical meaning. Therefore, your code must consistenly use ",[1031,88698,88699],{},"either"," tabs or spaces for indentation without deviation. The ",[49,88702,88704],{"href":88703},"https:\u002F\u002Fpeps.python.org\u002Fpep-0008\u002F","offical style guide for Python (“PEP 8”)"," specifies an indentation level as ",[154,88707,12152],{}," spaces wide.",[13,88710,88712],{"id":88711},"functions","Functions",[498,88714,88716],{"className":500,"code":88715,"language":502,"meta":104,"style":104},"def add(a, b):\n    return a + b\n",[504,88717,88718,88734],{"__ignoreMap":104},[507,88719,88720,88722,88724,88726,88728,88730,88732],{"class":509,"line":510},[507,88721,1370],{"class":513},[507,88723,87354],{"class":576},[507,88725,580],{"class":517},[507,88727,49],{"class":1382},[507,88729,622],{"class":517},[507,88731,13102],{"class":1382},[507,88733,1883],{"class":517},[507,88735,88736,88738,88741,88743],{"class":509,"line":105},[507,88737,2504],{"class":513},[507,88739,88740],{"class":517}," a ",[507,88742,2107],{"class":572},[507,88744,88745],{"class":517}," b\n",[18,88747,88748],{},"Python functions capture variables lexically (like JS closures), but the write rules differ, explained in scoping below. Default arguments are evaluated once (careful!):",[498,88750,88752],{"className":500,"code":88751,"language":502,"meta":104,"style":104},"def append_to_list(value, lst=[]):  # BAD\n    lst.append(value)\n    return lst\nUse:\ndef append_to_list(value, lst=None):\n    if lst is None:\n        lst = []\n    return lst\n",[504,88753,88754,88776,88786,88793,88798,88818,88831,88840],{"__ignoreMap":104},[507,88755,88756,88758,88761,88763,88765,88767,88770,88773],{"class":509,"line":510},[507,88757,1370],{"class":513},[507,88759,88760],{"class":576}," append_to_list",[507,88762,580],{"class":517},[507,88764,64110],{"class":1382},[507,88766,622],{"class":517},[507,88768,88769],{"class":1382},"lst",[507,88771,88772],{"class":517},"=[]):  ",[507,88774,88775],{"class":562},"# BAD\n",[507,88777,88778,88781,88783],{"class":509,"line":105},[507,88779,88780],{"class":517},"    lst.",[507,88782,1939],{"class":576},[507,88784,88785],{"class":517},"(value)\n",[507,88787,88788,88790],{"class":509,"line":540},[507,88789,2504],{"class":513},[507,88791,88792],{"class":517}," lst\n",[507,88794,88795],{"class":509,"line":553},[507,88796,88797],{"class":517},"Use:\n",[507,88799,88800,88802,88804,88806,88808,88810,88812,88814,88816],{"class":509,"line":559},[507,88801,1370],{"class":513},[507,88803,88760],{"class":576},[507,88805,580],{"class":517},[507,88807,64110],{"class":1382},[507,88809,622],{"class":517},[507,88811,88769],{"class":1382},[507,88813,573],{"class":517},[507,88815,56764],{"class":583},[507,88817,1883],{"class":517},[507,88819,88820,88822,88825,88827,88829],{"class":509,"line":566},[507,88821,1717],{"class":513},[507,88823,88824],{"class":517}," lst ",[507,88826,37008],{"class":513},[507,88828,57507],{"class":583},[507,88830,1728],{"class":517},[507,88832,88833,88836,88838],{"class":509,"line":590},[507,88834,88835],{"class":517},"        lst ",[507,88837,573],{"class":572},[507,88839,1910],{"class":517},[507,88841,88842,88844],{"class":509,"line":610},[507,88843,2504],{"class":513},[507,88845,88792],{"class":517},[13,88847,88849],{"id":88848},"classes-python-style-oop","Classes (Python-style OOP)",[498,88851,88853],{"className":500,"code":88852,"language":502,"meta":104,"style":104},"class Car:\n    def __init__(self, make):\n        self.make = make\n\n    def honk(self):\n        print(\"beep\")\n\nc = Car(\"Tesla\")\nc.honk()\nMethods need self explicitly. Inheritance uses the class name:\nclass SportsCar(Car):\n    pass\n",[504,88854,88855,88864,88884,88897,88901,88914,88925,88929,88944,88954,88971,88985],{"__ignoreMap":104},[507,88856,88857,88859,88862],{"class":509,"line":510},[507,88858,68362],{"class":513},[507,88860,88861],{"class":68365}," Car",[507,88863,1728],{"class":517},[507,88865,88866,88868,88871,88873,88877,88879,88882],{"class":509,"line":105},[507,88867,83071],{"class":513},[507,88869,88870],{"class":572}," __init__",[507,88872,580],{"class":517},[507,88874,88876],{"class":88875},"sKU4T","self",[507,88878,622],{"class":517},[507,88880,88881],{"class":1382},"make",[507,88883,1883],{"class":517},[507,88885,88886,88889,88892,88894],{"class":509,"line":540},[507,88887,88888],{"class":68365},"        self",[507,88890,88891],{"class":517},".make ",[507,88893,573],{"class":572},[507,88895,88896],{"class":517}," make\n",[507,88898,88899],{"class":509,"line":553},[507,88900,556],{"emptyLinePlaceholder":133},[507,88902,88903,88905,88908,88910,88912],{"class":509,"line":559},[507,88904,83071],{"class":513},[507,88906,88907],{"class":576}," honk",[507,88909,580],{"class":517},[507,88911,88876],{"class":88875},[507,88913,1883],{"class":517},[507,88915,88916,88918,88920,88923],{"class":509,"line":566},[507,88917,64185],{"class":572},[507,88919,580],{"class":517},[507,88921,88922],{"class":730},"\"beep\"",[507,88924,587],{"class":517},[507,88926,88927],{"class":509,"line":590},[507,88928,556],{"emptyLinePlaceholder":133},[507,88930,88931,88933,88935,88937,88939,88942],{"class":509,"line":610},[507,88932,58252],{"class":517},[507,88934,573],{"class":572},[507,88936,88861],{"class":576},[507,88938,580],{"class":517},[507,88940,88941],{"class":730},"\"Tesla\"",[507,88943,587],{"class":517},[507,88945,88946,88949,88952],{"class":509,"line":634},[507,88947,88948],{"class":517},"c.",[507,88950,88951],{"class":576},"honk",[507,88953,781],{"class":517},[507,88955,88956,88959,88961,88964,88966,88969],{"class":509,"line":661},[507,88957,88958],{"class":517},"Methods need ",[507,88960,88876],{"class":68365},[507,88962,88963],{"class":517}," explicitly. Inheritance uses the ",[507,88965,68362],{"class":513},[507,88967,88968],{"class":68365}," name",[507,88970,1728],{"class":517},[507,88972,88973,88975,88978,88980,88983],{"class":509,"line":678},[507,88974,68362],{"class":513},[507,88976,88977],{"class":68365}," SportsCar",[507,88979,580],{"class":517},[507,88981,88982],{"class":68365},"Car",[507,88984,1883],{"class":517},[507,88986,88987],{"class":509,"line":683},[507,88988,59443],{"class":513},[13,88990,88992],{"id":88991},"collections","Collections",[2513,88994,88996],{"id":88995},"list-array","List (array)",[498,88998,89000],{"className":500,"code":88999,"language":502,"meta":104,"style":104},"a = [1, 2, 3]\n",[504,89001,89002],{"__ignoreMap":104},[507,89003,89004,89007,89009,89011,89013,89015,89017,89019,89021],{"class":509,"line":510},[507,89005,89006],{"class":517},"a ",[507,89008,573],{"class":572},[507,89010,8427],{"class":517},[507,89012,625],{"class":583},[507,89014,622],{"class":517},[507,89016,584],{"class":583},[507,89018,622],{"class":517},[507,89020,8226],{"class":583},[507,89022,1794],{"class":517},[2513,89024,89026],{"id":89025},"tuple-immutable","Tuple (immutable)",[498,89028,89030],{"className":500,"code":89029,"language":502,"meta":104,"style":104},"t = (1, 2, 3)\n",[504,89031,89032],{"__ignoreMap":104},[507,89033,89034,89036,89038,89040,89042,89044,89046,89048,89050],{"class":509,"line":510},[507,89035,9307],{"class":517},[507,89037,573],{"class":572},[507,89039,58644],{"class":517},[507,89041,625],{"class":583},[507,89043,622],{"class":517},[507,89045,584],{"class":583},[507,89047,622],{"class":517},[507,89049,8226],{"class":583},[507,89051,587],{"class":517},[2513,89053,89055],{"id":89054},"dict","Dict",[498,89057,89059],{"className":500,"code":89058,"language":502,"meta":104,"style":104},"d = {\"name\": \"Stewart\", \"role\": \"CTO\"}\n",[504,89060,89061],{"__ignoreMap":104},[507,89062,89063,89066,89068,89070,89073,89075,89078,89080,89083,89085,89088],{"class":509,"line":510},[507,89064,89065],{"class":517},"d ",[507,89067,573],{"class":572},[507,89069,73290],{"class":517},[507,89071,89072],{"class":730},"\"name\"",[507,89074,1403],{"class":517},[507,89076,89077],{"class":730},"\"Stewart\"",[507,89079,622],{"class":517},[507,89081,89082],{"class":730},"\"role\"",[507,89084,1403],{"class":517},[507,89086,89087],{"class":730},"\"CTO\"",[507,89089,23875],{"class":517},[2513,89091,89092],{"id":1944},"Set",[498,89094,89096],{"className":500,"code":89095,"language":502,"meta":104,"style":104},"s = {1, 2, 3}\n",[504,89097,89098],{"__ignoreMap":104},[507,89099,89100,89103,89105,89107,89109,89111,89113,89115,89117],{"class":509,"line":510},[507,89101,89102],{"class":517},"s ",[507,89104,573],{"class":572},[507,89106,73290],{"class":517},[507,89108,625],{"class":583},[507,89110,622],{"class":517},[507,89112,584],{"class":583},[507,89114,622],{"class":517},[507,89116,8226],{"class":583},[507,89118,23875],{"class":517},[18,89120,89121],{},"List comprehensions:",[498,89123,89125],{"className":500,"code":89124,"language":502,"meta":104,"style":104},"squares = [x*x for x in range(10)]\n",[504,89126,89127],{"__ignoreMap":104},[507,89128,89129,89132,89134,89137,89139,89141,89143,89146,89148,89150,89152,89154],{"class":509,"line":510},[507,89130,89131],{"class":517},"squares ",[507,89133,573],{"class":572},[507,89135,89136],{"class":517}," [x",[507,89138,2391],{"class":572},[507,89140,69745],{"class":517},[507,89142,1630],{"class":513},[507,89144,89145],{"class":517}," x ",[507,89147,1636],{"class":513},[507,89149,8221],{"class":572},[507,89151,580],{"class":517},[507,89153,23805],{"class":583},[507,89155,82079],{"class":517},[18,89157,89158],{},"Dictionary comprehensions:",[498,89160,89162],{"className":500,"code":89161,"language":502,"meta":104,"style":104},"doubles = {x: x*2 for x in range(5)}\n",[504,89163,89164],{"__ignoreMap":104},[507,89165,89166,89169,89171,89174,89176,89178,89180,89182,89184,89186,89188,89190],{"class":509,"line":510},[507,89167,89168],{"class":517},"doubles ",[507,89170,573],{"class":572},[507,89172,89173],{"class":517}," {x: x",[507,89175,2391],{"class":572},[507,89177,584],{"class":583},[507,89179,8774],{"class":513},[507,89181,89145],{"class":517},[507,89183,1636],{"class":513},[507,89185,8221],{"class":572},[507,89187,580],{"class":517},[507,89189,58245],{"class":583},[507,89191,79662],{"class":517},[13,89193,89195],{"id":89194},"modules-imports","Modules \u002F Imports",[498,89197,89199],{"className":500,"code":89198,"language":502,"meta":104,"style":104},"import math\nfrom pathlib import Path\nfrom mymodule import helper\n",[504,89200,89201,89207,89219],{"__ignoreMap":104},[507,89202,89203,89205],{"class":509,"line":510},[507,89204,514],{"class":513},[507,89206,57135],{"class":517},[507,89208,89209,89211,89214,89216],{"class":509,"line":105},[507,89210,529],{"class":513},[507,89212,89213],{"class":517}," pathlib ",[507,89215,514],{"class":513},[507,89217,89218],{"class":517}," Path\n",[507,89220,89221,89223,89226,89228],{"class":509,"line":540},[507,89222,529],{"class":513},[507,89224,89225],{"class":517}," mymodule ",[507,89227,514],{"class":513},[507,89229,89230],{"class":517}," helper\n",[18,89232,89233,89234,53],{},"Scripts are modules. Packages are directories with ",[504,89235,89236],{},"__init__.py",[13,89238,89240],{"id":89239},"exceptions","Exceptions",[498,89242,89244],{"className":500,"code":89243,"language":502,"meta":104,"style":104},"try:\n    risky()\nexcept ValueError:\n    print(\"Nope\")\nexcept Exception as e:\n    print(\"Other error:\", e)\nfinally:\n    cleanup()\n",[504,89245,89246,89252,89259,89266,89277,89289,89301,89308],{"__ignoreMap":104},[507,89247,89248,89250],{"class":509,"line":510},[507,89249,57188],{"class":513},[507,89251,1728],{"class":517},[507,89253,89254,89257],{"class":509,"line":105},[507,89255,89256],{"class":576},"    risky",[507,89258,781],{"class":517},[507,89260,89261,89263],{"class":509,"line":540},[507,89262,57221],{"class":513},[507,89264,89265],{"class":517}," ValueError:\n",[507,89267,89268,89270,89272,89275],{"class":509,"line":553},[507,89269,2060],{"class":572},[507,89271,580],{"class":517},[507,89273,89274],{"class":730},"\"Nope\"",[507,89276,587],{"class":517},[507,89278,89279,89281,89284,89286],{"class":509,"line":559},[507,89280,57221],{"class":513},[507,89282,89283],{"class":517}," Exception ",[507,89285,521],{"class":513},[507,89287,89288],{"class":517}," e:\n",[507,89290,89291,89293,89295,89298],{"class":509,"line":566},[507,89292,2060],{"class":572},[507,89294,580],{"class":517},[507,89296,89297],{"class":730},"\"Other error:\"",[507,89299,89300],{"class":517},", e)\n",[507,89302,89303,89306],{"class":509,"line":590},[507,89304,89305],{"class":513},"finally",[507,89307,1728],{"class":517},[507,89309,89310,89313],{"class":509,"line":610},[507,89311,89312],{"class":576},"    cleanup",[507,89314,781],{"class":517},[13,89316,89318],{"id":89317},"file-io","File I\u002FO",[498,89320,89322],{"className":500,"code":89321,"language":502,"meta":104,"style":104},"with open(\"data.txt\") as f:\n    text = f.read()\n",[504,89323,89324,89341],{"__ignoreMap":104},[507,89325,89326,89328,89330,89332,89335,89337,89339],{"class":509,"line":510},[507,89327,69626],{"class":513},[507,89329,69629],{"class":572},[507,89331,580],{"class":517},[507,89333,89334],{"class":730},"\"data.txt\"",[507,89336,655],{"class":517},[507,89338,521],{"class":513},[507,89340,69646],{"class":517},[507,89342,89343,89346,89348,89351,89354],{"class":509,"line":105},[507,89344,89345],{"class":517},"    text ",[507,89347,573],{"class":572},[507,89349,89350],{"class":517}," f.",[507,89352,89353],{"class":576},"read",[507,89355,781],{"class":517},[18,89357,54828,89358,89360],{},[504,89359,69626],{}," block ensures cleanup (context manager).",[13,89362,89364],{"id":89363},"python-variable-scoping-legb-rule","Python Variable Scoping (LEGB Rule)",[18,89366,89367],{},"Python scoping follows L-E-G-B: - Local, inside current function - Enclosing, outer function scopes (closures) - Global, module-level - Built-in, len, print, etc This is very similar to JavaScript lexical scoping except Python treats assignment differently.",[2513,89369,89371],{"id":89370},"important-rule","Important Rule",[18,89373,89374],{},"Any variable you assign to inside a function is considered local unless you explicitly declare otherwise. This catches people:",[498,89376,89378],{"className":500,"code":89377,"language":502,"meta":104,"style":104},"x = 10\n\ndef f():\n    print(x)     # ERROR? Actually, UnboundLocalError!\n    x = 20\n",[504,89379,89380,89388,89392,89400,89410],{"__ignoreMap":104},[507,89381,89382,89384,89386],{"class":509,"line":510},[507,89383,69745],{"class":517},[507,89385,573],{"class":572},[507,89387,88615],{"class":583},[507,89389,89390],{"class":509,"line":105},[507,89391,556],{"emptyLinePlaceholder":133},[507,89393,89394,89396,89398],{"class":509,"line":540},[507,89395,1370],{"class":513},[507,89397,64432],{"class":576},[507,89399,1930],{"class":517},[507,89401,89402,89404,89407],{"class":509,"line":553},[507,89403,2060],{"class":572},[507,89405,89406],{"class":517},"(x)     ",[507,89408,89409],{"class":562},"# ERROR? Actually, UnboundLocalError!\n",[507,89411,89412,89415,89417],{"class":509,"line":559},[507,89413,89414],{"class":517},"    x ",[507,89416,573],{"class":572},[507,89418,89419],{"class":583}," 20\n",[18,89421,89422,89423,89426,89427,89429,89430,89432],{},"Why? Because the presence of ",[504,89424,89425],{},"x = 20"," makes ",[504,89428,9139],{}," local to ",[504,89431,22278],{},". Python sees “you assign to x somewhere in the function” → therefore x is local everywhere in that function. To mutate the module-level x:",[498,89434,89436],{"className":500,"code":89435,"language":502,"meta":104,"style":104},"x = 10\n\ndef f():\n    global x\n    print(x)\n    x = 20\n",[504,89437,89438,89446,89450,89458,89466,89473],{"__ignoreMap":104},[507,89439,89440,89442,89444],{"class":509,"line":510},[507,89441,69745],{"class":517},[507,89443,573],{"class":572},[507,89445,88615],{"class":583},[507,89447,89448],{"class":509,"line":105},[507,89449,556],{"emptyLinePlaceholder":133},[507,89451,89452,89454,89456],{"class":509,"line":540},[507,89453,1370],{"class":513},[507,89455,64432],{"class":576},[507,89457,1930],{"class":517},[507,89459,89460,89463],{"class":509,"line":553},[507,89461,89462],{"class":513},"    global",[507,89464,89465],{"class":517}," x\n",[507,89467,89468,89470],{"class":509,"line":559},[507,89469,2060],{"class":572},[507,89471,89472],{"class":517},"(x)\n",[507,89474,89475,89477,89479],{"class":509,"line":566},[507,89476,89414],{"class":517},[507,89478,573],{"class":572},[507,89480,89419],{"class":583},[2513,89482,89484],{"id":89483},"closures-enclosing-scope","Closures (Enclosing scope)",[498,89486,89488],{"className":500,"code":89487,"language":502,"meta":104,"style":104},"def outer():\n    x = 10\n    def inner():\n        print(x)     # OK\n    inner()\n",[504,89489,89490,89499,89507,89516,89525],{"__ignoreMap":104},[507,89491,89492,89494,89497],{"class":509,"line":510},[507,89493,1370],{"class":513},[507,89495,89496],{"class":576}," outer",[507,89498,1930],{"class":517},[507,89500,89501,89503,89505],{"class":509,"line":105},[507,89502,89414],{"class":517},[507,89504,573],{"class":572},[507,89506,88615],{"class":583},[507,89508,89509,89511,89514],{"class":509,"line":540},[507,89510,83071],{"class":513},[507,89512,89513],{"class":576}," inner",[507,89515,1930],{"class":517},[507,89517,89518,89520,89522],{"class":509,"line":553},[507,89519,64185],{"class":572},[507,89521,89406],{"class":517},[507,89523,89524],{"class":562},"# OK\n",[507,89526,89527,89530],{"class":509,"line":559},[507,89528,89529],{"class":576},"    inner",[507,89531,781],{"class":517},[18,89533,89534],{},"But trying to assign inside the closure fails:",[498,89536,89538],{"className":500,"code":89537,"language":502,"meta":104,"style":104},"def outer():\n    x = 10\n    def inner():\n        x = 20      # This creates NEW local x\n    inner()\n    print(x)        # still 10\n",[504,89539,89540,89548,89556,89564,89576,89582],{"__ignoreMap":104},[507,89541,89542,89544,89546],{"class":509,"line":510},[507,89543,1370],{"class":513},[507,89545,89496],{"class":576},[507,89547,1930],{"class":517},[507,89549,89550,89552,89554],{"class":509,"line":105},[507,89551,89414],{"class":517},[507,89553,573],{"class":572},[507,89555,88615],{"class":583},[507,89557,89558,89560,89562],{"class":509,"line":540},[507,89559,83071],{"class":513},[507,89561,89513],{"class":576},[507,89563,1930],{"class":517},[507,89565,89566,89568,89570,89573],{"class":509,"line":553},[507,89567,77298],{"class":517},[507,89569,573],{"class":572},[507,89571,89572],{"class":583}," 20",[507,89574,89575],{"class":562},"      # This creates NEW local x\n",[507,89577,89578,89580],{"class":509,"line":559},[507,89579,89529],{"class":576},[507,89581,781],{"class":517},[507,89583,89584,89586,89589],{"class":509,"line":566},[507,89585,2060],{"class":572},[507,89587,89588],{"class":517},"(x)        ",[507,89590,89591],{"class":562},"# still 10\n",[18,89593,89594],{},"To mutate the enclosing scope’s variable use nonlocal:",[498,89596,89598],{"className":500,"code":89597,"language":502,"meta":104,"style":104},"def outer():\n    x = 10\n    def inner():\n        nonlocal x\n        x = 20\n    inner()\n    print(x)   # 20\n",[504,89599,89600,89608,89616,89624,89631,89639,89645],{"__ignoreMap":104},[507,89601,89602,89604,89606],{"class":509,"line":510},[507,89603,1370],{"class":513},[507,89605,89496],{"class":576},[507,89607,1930],{"class":517},[507,89609,89610,89612,89614],{"class":509,"line":105},[507,89611,89414],{"class":517},[507,89613,573],{"class":572},[507,89615,88615],{"class":583},[507,89617,89618,89620,89622],{"class":509,"line":540},[507,89619,83071],{"class":513},[507,89621,89513],{"class":576},[507,89623,1930],{"class":517},[507,89625,89626,89629],{"class":509,"line":553},[507,89627,89628],{"class":513},"        nonlocal",[507,89630,89465],{"class":517},[507,89632,89633,89635,89637],{"class":509,"line":559},[507,89634,77298],{"class":517},[507,89636,573],{"class":572},[507,89638,89419],{"class":583},[507,89640,89641,89643],{"class":509,"line":566},[507,89642,89529],{"class":576},[507,89644,781],{"class":517},[507,89646,89647,89649,89652],{"class":509,"line":590},[507,89648,2060],{"class":572},[507,89650,89651],{"class":517},"(x)   ",[507,89653,89654],{"class":562},"# 20\n",[13,89656,89658],{"id":89657},"pythonic-style-cheat-sheet","Pythonic Style Cheat Sheet",[18,89660,89661],{},"Idiomatic ways to write things:",[2513,89663,89665],{"id":89664},"iteration","Iteration",[498,89667,89669],{"className":500,"code":89668,"language":502,"meta":104,"style":104},"for x in items:\n    ...\n",[504,89670,89671,89682],{"__ignoreMap":104},[507,89672,89673,89675,89677,89679],{"class":509,"line":510},[507,89674,1630],{"class":513},[507,89676,89145],{"class":517},[507,89678,1636],{"class":513},[507,89680,89681],{"class":517}," items:\n",[507,89683,89684],{"class":509,"line":105},[507,89685,89686],{"class":583},"    ...\n",[2513,89688,89690],{"id":89689},"enumerate-with-index","Enumerate with index",[498,89692,89694],{"className":500,"code":89693,"language":502,"meta":104,"style":104},"for i, x in enumerate(items):\n    ...\n",[504,89695,89696,89710],{"__ignoreMap":104},[507,89697,89698,89700,89703,89705,89707],{"class":509,"line":510},[507,89699,1630],{"class":513},[507,89701,89702],{"class":517}," i, x ",[507,89704,1636],{"class":513},[507,89706,1957],{"class":572},[507,89708,89709],{"class":517},"(items):\n",[507,89711,89712],{"class":509,"line":105},[507,89713,89686],{"class":583},[2513,89715,89717],{"id":89716},"iterate-dict","Iterate dict",[498,89719,89721],{"className":500,"code":89720,"language":502,"meta":104,"style":104},"for k, v in d.items():\n    ...\n",[504,89722,89723,89739],{"__ignoreMap":104},[507,89724,89725,89727,89730,89732,89735,89737],{"class":509,"line":510},[507,89726,1630],{"class":513},[507,89728,89729],{"class":517}," k, v ",[507,89731,1636],{"class":513},[507,89733,89734],{"class":517}," d.",[507,89736,22607],{"class":576},[507,89738,1930],{"class":517},[507,89740,89741],{"class":509,"line":105},[507,89742,89686],{"class":583},[2513,89744,89746],{"id":89745},"ternary","Ternary",[498,89748,89750],{"className":500,"code":89749,"language":502,"meta":104,"style":104},"msg = \"ok\" if status else \"bad\"\n",[504,89751,89752],{"__ignoreMap":104},[507,89753,89754,89757,89759,89762,89764,89766,89768],{"class":509,"line":510},[507,89755,89756],{"class":517},"msg ",[507,89758,573],{"class":572},[507,89760,89761],{"class":730}," \"ok\"",[507,89763,66162],{"class":513},[507,89765,81435],{"class":517},[507,89767,61407],{"class":513},[507,89769,89770],{"class":730}," \"bad\"\n",[2513,89772,89774],{"id":89773},"with-open-file-handling","With-open file handling",[498,89776,89778],{"className":500,"code":89777,"language":502,"meta":104,"style":104},"with open(\"file.txt\") as f:\n    data = f.read()\n",[504,89779,89780,89797],{"__ignoreMap":104},[507,89781,89782,89784,89786,89788,89791,89793,89795],{"class":509,"line":510},[507,89783,69626],{"class":513},[507,89785,69629],{"class":572},[507,89787,580],{"class":517},[507,89789,89790],{"class":730},"\"file.txt\"",[507,89792,655],{"class":517},[507,89794,521],{"class":513},[507,89796,69646],{"class":517},[507,89798,89799,89802,89804,89806,89808],{"class":509,"line":105},[507,89800,89801],{"class":517},"    data ",[507,89803,573],{"class":572},[507,89805,89350],{"class":517},[507,89807,89353],{"class":576},[507,89809,781],{"class":517},[2513,89811,89813],{"id":89812},"write-generator-expression","Write generator expression",[498,89815,89817],{"className":500,"code":89816,"language":502,"meta":104,"style":104},"total = sum(x*x for x in nums)\n",[504,89818,89819],{"__ignoreMap":104},[507,89820,89821,89823,89825,89827,89830,89832,89834,89836,89838,89840],{"class":509,"line":510},[507,89822,78935],{"class":517},[507,89824,573],{"class":572},[507,89826,8815],{"class":572},[507,89828,89829],{"class":517},"(x",[507,89831,2391],{"class":572},[507,89833,69745],{"class":517},[507,89835,1630],{"class":513},[507,89837,89145],{"class":517},[507,89839,1636],{"class":513},[507,89841,89842],{"class":517}," nums)\n",[13,89844,89846],{"id":89845},"tiny-gotchas-you-should-know-immediately","Tiny “gotchas” you should know immediately",[18,89848,89849],{},"Mutability matters. Lists, dicts, sets = mutable. Tuples, ints, strings = immutable. Everything is reference by default.",[498,89851,89853],{"className":500,"code":89852,"language":502,"meta":104,"style":104},"a = [1,2]\nb = a\nb.append(3)\na == [1,2,3]\u002F\u002F True!\n\n",[504,89854,89855,89871,89881,89894],{"__ignoreMap":104},[507,89856,89857,89859,89861,89863,89865,89867,89869],{"class":509,"line":510},[507,89858,89006],{"class":517},[507,89860,573],{"class":572},[507,89862,8427],{"class":517},[507,89864,625],{"class":583},[507,89866,2819],{"class":517},[507,89868,584],{"class":583},[507,89870,1794],{"class":517},[507,89872,89873,89876,89878],{"class":509,"line":105},[507,89874,89875],{"class":517},"b ",[507,89877,573],{"class":572},[507,89879,89880],{"class":517}," a\n",[507,89882,89883,89886,89888,89890,89892],{"class":509,"line":540},[507,89884,89885],{"class":517},"b.",[507,89887,1939],{"class":576},[507,89889,580],{"class":517},[507,89891,8226],{"class":583},[507,89893,587],{"class":517},[507,89895,89896,89898,89900,89902,89904,89906,89908,89910,89912,89914,89917,89919],{"class":509,"line":553},[507,89897,89006],{"class":517},[507,89899,1723],{"class":572},[507,89901,8427],{"class":517},[507,89903,625],{"class":583},[507,89905,2819],{"class":517},[507,89907,584],{"class":583},[507,89909,2819],{"class":517},[507,89911,8226],{"class":583},[507,89913,12273],{"class":517},[507,89915,89916],{"class":572},"\u002F\u002F",[507,89918,64764],{"class":583},[507,89920,89921],{"class":517},"!\n",[13,89923,89925],{"id":89924},"truthiness","Truthiness:",[498,89927,89929],{"className":500,"code":89928,"language":502,"meta":104,"style":104},"0, \"\", [], {}, None, False → falsey\n",[504,89930,89931],{"__ignoreMap":104},[507,89932,89933,89935,89937,89939,89942,89944,89946,89948],{"class":509,"line":510},[507,89934,601],{"class":583},[507,89936,622],{"class":517},[507,89938,8430],{"class":730},[507,89940,89941],{"class":517},", [], {}, ",[507,89943,56764],{"class":583},[507,89945,622],{"class":517},[507,89947,21964],{"class":583},[507,89949,89950],{"class":517}," → falsey\n",[18,89952,89953],{},"Everything else is truthy.",[13,89955,89957],{"id":89956},"_10-line-example-that-uses-everything-above","10-line example that uses everything above",[498,89959,89961],{"className":500,"code":89960,"language":502,"meta":104,"style":104},"def make_counter():\n    count = 0\n    def inc():\n        nonlocal count\n        count += 1\n        return count\n    return inc\n\ncounter = make_counter()\nprint(counter())  # 1\nprint(counter())  # 2\n",[504,89962,89963,89972,89981,89990,89997,90006,90012,90019,90023,90034,90049],{"__ignoreMap":104},[507,89964,89965,89967,89970],{"class":509,"line":510},[507,89966,1370],{"class":513},[507,89968,89969],{"class":576}," make_counter",[507,89971,1930],{"class":517},[507,89973,89974,89977,89979],{"class":509,"line":105},[507,89975,89976],{"class":517},"    count ",[507,89978,573],{"class":572},[507,89980,2246],{"class":583},[507,89982,89983,89985,89988],{"class":509,"line":540},[507,89984,83071],{"class":513},[507,89986,89987],{"class":576}," inc",[507,89989,1930],{"class":517},[507,89991,89992,89994],{"class":509,"line":553},[507,89993,89628],{"class":513},[507,89995,89996],{"class":517}," count\n",[507,89998,89999,90002,90004],{"class":509,"line":559},[507,90000,90001],{"class":517},"        count ",[507,90003,2285],{"class":572},[507,90005,2084],{"class":583},[507,90007,90008,90010],{"class":509,"line":566},[507,90009,64873],{"class":513},[507,90011,89996],{"class":517},[507,90013,90014,90016],{"class":509,"line":590},[507,90015,2504],{"class":513},[507,90017,90018],{"class":517}," inc\n",[507,90020,90021],{"class":509,"line":610},[507,90022,556],{"emptyLinePlaceholder":133},[507,90024,90025,90028,90030,90032],{"class":509,"line":634},[507,90026,90027],{"class":517},"counter ",[507,90029,573],{"class":572},[507,90031,89969],{"class":576},[507,90033,781],{"class":517},[507,90035,90036,90038,90040,90043,90046],{"class":509,"line":661},[507,90037,8525],{"class":572},[507,90039,580],{"class":517},[507,90041,90042],{"class":576},"counter",[507,90044,90045],{"class":517},"())  ",[507,90047,90048],{"class":562},"# 1\n",[507,90050,90051,90053,90055,90057,90059],{"class":509,"line":678},[507,90052,8525],{"class":572},[507,90054,580],{"class":517},[507,90056,90042],{"class":576},[507,90058,90045],{"class":517},[507,90060,90061],{"class":562},"# 2\n",[953,90063,90064],{},"html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sKU4T, html code.shiki .sKU4T{--shiki-default:#E5C07B;--shiki-default-font-style:italic}",{"title":104,"searchDepth":105,"depth":105,"links":90066},[90067,90068,90069,90070,90071,90077,90078,90079,90080,90084,90092,90093,90094],{"id":88485,"depth":105,"text":88486},{"id":88585,"depth":105,"text":88586},{"id":88711,"depth":105,"text":88712},{"id":88848,"depth":105,"text":88849},{"id":88991,"depth":105,"text":88992,"children":90072},[90073,90074,90075,90076],{"id":88995,"depth":540,"text":88996},{"id":89025,"depth":540,"text":89026},{"id":89054,"depth":540,"text":89055},{"id":1944,"depth":540,"text":89092},{"id":89194,"depth":105,"text":89195},{"id":89239,"depth":105,"text":89240},{"id":89317,"depth":105,"text":89318},{"id":89363,"depth":105,"text":89364,"children":90081},[90082,90083],{"id":89370,"depth":540,"text":89371},{"id":89483,"depth":540,"text":89484},{"id":89657,"depth":105,"text":89658,"children":90085},[90086,90087,90088,90089,90090,90091],{"id":89664,"depth":540,"text":89665},{"id":89689,"depth":540,"text":89690},{"id":89716,"depth":540,"text":89717},{"id":89745,"depth":540,"text":89746},{"id":89773,"depth":540,"text":89774},{"id":89812,"depth":540,"text":89813},{"id":89845,"depth":105,"text":89846},{"id":89924,"depth":105,"text":89925},{"id":89956,"depth":105,"text":89957},[112,85993,85994,88459],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Python has become the de facto programming language of quantum computing. This guide highlights some basic Python commands and logic; foundational elements to build from.","Python has become the de facto language of quantum computing. This guide covers the basic Python commands and logic to build from.",{"image":90101,"alt":88459},"\u002F_content\u002Fimages\u002Fpython-basics\u002Fhero.webp",{},{"slug":90104,"title":90105,"desc":90106},"\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fqiskit-setup","3 · Setup IBM’s Qiskit SDK","This guide details how to install IBM’s Qiskit quantum computing packages for Python, using the UV package manager.","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-basics","5 min read",[],{"title":90111,"description":90099},"Learn Python language basics · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-basics",[],"Zs0gGrUcRe99ICeSFSxgdwYG6Yjp2VxvMrvwkeb6PD8",{"id":90116,"title":88481,"authors":90117,"body":90118,"breadcrumb":91045,"builders":91046,"byline":91047,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":88428,"description":88428,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":91048,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":91050,"navigation":133,"newsItems":116,"next":91051,"ogImage":116,"order":510,"outcomes":116,"path":91054,"publishDate":88450,"readingTime":91055,"related":91056,"relatedProjects":116,"seo":91057,"stem":91059,"tags":91060,"track":86016,"trackName":85994,"__hash__":91061},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-setup.md",[86363],{"type":10,"value":90119,"toc":91020},[90120,90157,90161,90182,90186,90203,90207,90233,90266,90270,90304,90308,90325,90331,90335,90353,90369,90373,90380,90395,90415,90419,90426,90430,90444,90450,90462,90468,90477,90480,90484,90487,90501,90505,90508,90511,90515,90518,90532,90540,90544,90547,90565,90578,90582,90585,90603,90609,90613,90616,90620,90623,90630,90642,90646,90649,90654,90666,90670,90677,90680,90691,90703,90707,90713,90725,90728,90765,90768,90797,90801,90804,90818,90821,90837,90843,90847,90886,90899,90996,90998,91001,91017],[18,90121,90122,90123,90126,90127,90130,90131,90133,90134,90136,90137,90140,90141,90144,90145,90148,90149,90152,90153,90156],{},"Like any language, ",[49,90124,496],{"href":90125},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPython_(programming_language)"," is a collection of ",[1031,90128,90129],{},"ideas"," for how to communicate. We can’t download and install ",[1031,90132,90129],{},". But we can download and install a ",[1031,90135,81072],{}," that follows these ideas to the letter; using them as a strict guide for translating our commands into action. This sort of program is called an ",[1031,90138,90139],{},"interpreter."," For brevity we’re going to use “Python” interchangeably to refer to both the ",[1031,90142,90143],{},"language (ideas)"," and the actual ",[1031,90146,90147],{},"interpreter software"," that puts those ideas into action. When you encounter phrases like “running Python” you’ll know this actually means running a Python ",[1031,90150,90151],{},"interpreter"," to make use of the Python ",[1031,90154,90155],{},"language."," Let’s get to it.",[13,90158,90160],{"id":90159},"juggling-pythons","Juggling pythons",[18,90162,90163,90164,90167,90168,90172,90173,86110,90177,90181],{},"There are many versions of Python. Perhaps your computer shipped with a particular version of Python pre-installed by the manufacturer. This is referred to as your “System Python” because it’s a specific version of Python used by your machine’s ",[1031,90165,90166],{},"operating system",", like ",[49,90169,90171],{"href":90170},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMacOS","Apple’s macOS",", or a brand of ",[49,90174,90176],{"href":90175},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLinux","Linux",[49,90178,90180],{"href":90179},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMicrosoft_Windows","Microsoft’s Windows",". Because your OS relies on its System Python remaining exactly in its current state (with its specific version, settings, plugins, and so on), we’d like to leave it untouched. Thankfully, it’s easy to install multiple versions of Python alongside each other, keeping them completely separate from one another. (Installing multiple versions of Python is a common practice, and it doesn’t take up much hard drive space.)",[13,90183,90185],{"id":90184},"_1-open-a-shell","1. Open a “shell”",[18,90187,90188,90189,90193,90194,90198,90199,53],{},"A “shell” is a program that allows you to enter text-based commands through a ",[49,90190,90192],{"href":90191},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCommand-line_interface","command-line interface"," (or “CLI” for short). This is in contrast to a ",[49,90195,90197],{"href":90196},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGraphical_user_interface","graphic user interface"," (or “GUI”) that you might be accustomed to; using a pointing device like a mouse or your finger to interact with graphic representations of data and actions to perform. (The term “shell” is a linguistic expansion on labeling an operating system’s core as its “kernel.” The shell “wraps” the kernel and is the user-facing surface that handles interactions with it.) For our purposes it’s unnecessary to become a shell expert, but if you’re curious to know more, this video is an excellent resource: ",[49,90200,90202],{"href":90201},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=IYZDIhfAUM0","Become a shell wizard in ~12 mins",[2513,90204,90206],{"id":90205},"open-a-shell-in-macos","Open a shell in MacOS",[27370,90208,90209,90220,90226],{},[45,90210,90211,90212,90215,90216,90219],{},"Press your keyboard’s ",[154,90213,90214],{},"⌘"," key and ",[154,90217,90218],{},"spacebar"," simultaneously to open the Spotlight prompt.",[45,90221,90222,90223,53],{},"Type ",[504,90224,90225],{},"Terminal",[45,90227,90228,90229,90232],{},"Press the ",[154,90230,90231],{},"Enter"," key.",[18,90234,90235,90236,90240,90241,90245,90246,90250,90251,90255,90256,90259,90260,90262,90263,53],{},"This will open the ",[49,90237,90239],{"href":90238},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTerminal_(macOS)","macOS Terminal application",". On ",[49,90242,90244],{"href":90243},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMacOS_Ventura","macOS Ventura"," or later, Terminal defaults to ",[49,90247,90249],{"href":90248},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FZ_shell","Z Shell (zsh)",". On older versions of macOS, Terminal defaults to ",[49,90252,90254],{"href":90253},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBash%3C\u002Fem%3E(Unix_shell)","Bash Shell",". (Either shell is perfectly usable.) You can check which shell you are using by typing (or pasting in) ",[504,90257,90258],{},"echo $SHELL"," and pressing your ",[154,90261,90231],{}," key. This command reveals the file location of the shell program you are using, and the names within that location’s path will imply which shell is active. For example, if using Z Shell you might see the path ",[504,90264,90265],{},"\u002Fbin\u002Fzsh",[2513,90267,90269],{"id":90268},"open-a-shell-in-linux","Open a shell in Linux",[27370,90271,90272],{},[45,90273,90211,90274,622,90277,86508,90280,90282,90283,622,90287,622,90291,90295,90296,90300,90301,53],{},[154,90275,90276],{},"Ctrl",[154,90278,90279],{},"Alt",[154,90281,37046],{}," keys simultaneously to open the Terminal application. (This should work on ",[49,90284,90286],{"href":90285},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FUbuntu","Ubuntu",[49,90288,90290],{"href":90289},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FLinux_Mint","Linux Mint",[49,90292,90294],{"href":90293},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPop!_OS","Pop!_OS",", and many ",[49,90297,90299],{"href":90298},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGNOME","GNOME","-based Linux flavors.) The default shell is usually ",[49,90302,90303],{"href":90253},"Bash",[2513,90305,90307],{"id":90306},"open-a-shell-in-windows","Open a shell in Windows",[27370,90309,90310,90315,90321],{},[45,90311,90211,90312,90232],{},[154,90313,90314],{},"Windows",[45,90316,90222,90317,90320],{},[504,90318,90319],{},"PowerShell"," into the prompt area.",[45,90322,90228,90323,90232],{},[154,90324,90231],{},[18,90326,90235,90327,53],{},[49,90328,90330],{"href":90329},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPowerShell","Windows PowerShell application",[13,90332,90334],{"id":90333},"_2-install-uv","2. Install UV",[18,90336,90337,90338,947,90342,90345,90346,90348,90349,53],{},"We need a clean way to install a new version of Python on our system, and to keep it separate from any existing (or future) installations of Python. In the past we may have recommended solutions like ",[49,90339,90341],{"href":90340},"https:\u002F\u002Fwiki.python.org\u002Fmoin\u002FVirtualenv","virtualenv",[49,90343,87267],{"href":90344},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FConda%3C\u002Fem%3E(package_manager)",". But these days the winning solution is ",[49,90347,87272],{"href":87271},", a single application that is blazingly fast and replaces (or wraps) several common Python-related tools and package managers such as ",[49,90350,90352],{"href":90351},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPip%3C\u002Fem%3E(package_manager)","pip",[18,90354,90355,90356,90360,90361,90364,90365,53],{},"The following installation instructions come from ",[49,90357,90359],{"href":90358},"https:\u002F\u002Fdocs.astral.sh\u002Fuv\u002Fgetting-started\u002Finstallation\u002F","UV’s installation guide",". For troubleshooting or additional details, refer to their documentation: ",[49,90362,90363],{"href":90358},"https:\u002F\u002Fdocs.astral.sh\u002Fuv\u002Fgetting-started\u002Finstallation",". See also, the ",[49,90366,90368],{"href":90367},"https:\u002F\u002Fgithub.com\u002Fastral-sh\u002Fuv","UV GitHub repository",[2513,90370,90372],{"id":90371},"install-uv-on-macos-or-linux","Install UV on macOS or Linux",[18,90374,90375,90376,90379],{},"Enter the following command into your shell application: curl -LsSf ",[49,90377,90378],{"href":90378},"https:\u002F\u002Fastral.sh\u002Fuv\u002Finstall.sh"," | sh",[18,90381,90382,90383,90386,90387,90390,90391,90394],{},"If you receive an error similar to “",[504,90384,90385],{},"command not found: curl","”, it could mean the your system does not have ",[504,90388,90389],{},"curl"," installed. Don’t worry. Give this ",[504,90392,90393],{},"wget"," command a try instead:",[498,90396,90398],{"className":86859,"code":90397,"language":86861,"meta":104,"style":104},"wget -qO- https:\u002F\u002Fastral.sh\u002Fuv\u002Finstall.sh | sh\n",[504,90399,90400],{"__ignoreMap":104},[507,90401,90402,90404,90407,90410,90412],{"class":509,"line":510},[507,90403,90393],{"class":576},[507,90405,90406],{"class":583}," -qO-",[507,90408,90409],{"class":730}," https:\u002F\u002Fastral.sh\u002Fuv\u002Finstall.sh",[507,90411,87615],{"class":517},[507,90413,90414],{"class":576},"sh\n",[2513,90416,90418],{"id":90417},"install-uv-on-windows","Install UV on Windows",[18,90420,90421,90422,90425],{},"Enter the following command into your shell application: powershell -ExecutionPolicy ByPass -c \"irm ",[49,90423,90424],{"href":90424},"https:\u002F\u002Fastral.sh\u002Fuv\u002Finstall.ps1"," | iex\"",[2513,90427,90429],{"id":90428},"make-uv-available","Make UV available",[18,90431,90432,90433,90435,90436,90439,90440,90443],{},"Congratulations. You now have UV installed on your system. But the ",[504,90434,86898],{}," command is ",[1031,90437,90438],{},"not"," available in our current shell window. The easiest way to start using UV immediately is to ",[154,90441,90442],{},"close your current shell window and open a fresh new one",". Give that a try now.",[18,90445,90446,90447,90449],{},"In a fresh shell window, enter the following to ask for the UV version number. This will confirm that the ",[504,90448,86898],{}," command is available to you.",[498,90451,90453],{"className":86859,"code":90452,"language":86861,"meta":104,"style":104},"uv --version\n",[504,90454,90455],{"__ignoreMap":104},[507,90456,90457,90459],{"class":509,"line":510},[507,90458,86898],{"class":576},[507,90460,90461],{"class":583}," --version\n",[18,90463,90464,90465,90467],{},"Your shell should respond with one line indicating the freshly installed version number. (Did something not go as planned? That’s ok. Refer to ",[49,90466,90359],{"href":90358}," as a first step for troubleshooting.) For a full list of UV’s available commands, enter the following into your shell:",[498,90469,90471],{"className":86859,"code":90470,"language":86861,"meta":104,"style":104},"uv\n",[504,90472,90473],{"__ignoreMap":104},[507,90474,90475],{"class":509,"line":510},[507,90476,90470],{"class":576},[18,90478,90479],{},"That’s right, just asking for UV will provide you with a whole menu of commands and useful information.",[13,90481,90483],{"id":90482},"_3-install-python","3. Install Python",[18,90485,90486],{},"Using UV, we can now safely install a sandboxed Python, separate from any past or future installations of Python. To install the latest Python, enter the following into your shell:",[498,90488,90490],{"className":86859,"code":90489,"language":86861,"meta":104,"style":104},"uv python install\n",[504,90491,90492],{"__ignoreMap":104},[507,90493,90494,90496,90498],{"class":509,"line":510},[507,90495,86898],{"class":576},[507,90497,86904],{"class":730},[507,90499,90500],{"class":730}," install\n",[2513,90502,90504],{"id":90503},"macos-command-line-developer-tools","macOS command-line developer tools",[18,90506,90507],{},"If you’re on macOS and have not previously installed the command-line developer tools, you will be prompted to do so now. It’s a hefty download, but will enable you to compile and run all of the tools you may wish to use in the future, including installing Python right now.",[831,90509],{"alt":104,"caption":104,"no":104,"src":90510},"\u002F_content\u002Fimages\u002Fpython-setup\u002Fdeveloper-tools-2.webp",[2513,90512,90514],{"id":90513},"confirm-python-installed","Confirm Python installed",[18,90516,90517],{},"Once UV has finished installing the latest version of Python, you can verify this (as well as detect previously installed versions of Python) by entering the following into your shell:",[498,90519,90521],{"className":86859,"code":90520,"language":86861,"meta":104,"style":104},"uv python list\n",[504,90522,90523],{"__ignoreMap":104},[507,90524,90525,90527,90529],{"class":509,"line":510},[507,90526,86898],{"class":576},[507,90528,86904],{"class":730},[507,90530,90531],{"class":730}," list\n",[87238,90533,90534],{},[18,90535,90536,90537],{},"UV can do a whole lot more than list some versions of Python. Here’s a handy cheat sheet for UV’s management commands: ",[49,90538,90539],{"href":90539},"https:\u002F\u002Fdocs.astral.sh\u002Fuv\u002Fgetting-started\u002Ffeatures\u002F",[2513,90541,90543],{"id":90542},"hello-world","Hello, World!",[18,90545,90546],{},"Now that we definitely have Python installed, let’s run a tiny “Hello, World!” program right from the command line. Enter this into your shell:",[498,90548,90550],{"className":86859,"code":90549,"language":86861,"meta":104,"style":104},"uv run python -c 'print( \"Hello, World!\" )'\n",[504,90551,90552],{"__ignoreMap":104},[507,90553,90554,90556,90558,90560,90562],{"class":509,"line":510},[507,90555,86898],{"class":576},[507,90557,86901],{"class":730},[507,90559,86904],{"class":730},[507,90561,87379],{"class":583},[507,90563,90564],{"class":730}," 'print( \"Hello, World!\" )'\n",[18,90566,90567,90568,90570,90571,90574,90575,90577],{},"Our shell responds with “",[504,90569,90543],{},"” (And we can deduce that the ",[504,90572,90573],{},"-c"," flag tells ",[504,90576,86898],{}," to execute any command that follows.) This is progress. But running one command at a time isn’t going to get us very far. Let’s start thinking a little larger.",[13,90579,90581],{"id":90580},"_4-create-your-first-project","4. Create your first project",[18,90583,90584],{},"We’d like to create a sandboxed environment that both “pins” a specific version of Python for use, and houses any additional code packages we require. This ensures that anything we add to Python remains tidily within our sandbox, and anything done outside of our sandbox is kept at a safe distance and won’t damage our project. Our UV project will be composed of:",[42,90586,90587,90590,90597,90600],{},[45,90588,90589],{},"A folder.",[45,90591,90592,90593,90596],{},"A ",[504,90594,90595],{},"pyproject.toml"," file.",[45,90598,90599],{},"A local virtual environment (created automatically).",[45,90601,90602],{},"Our Python files.",[18,90604,90605,90606,90608],{},"Create a project folder on your Desktop called ",[504,90607,85760],{},". (The exact name and location of this folder doesn’t matter so much, as long as it’s easy for you to get to and work with.)",[2513,90610,90612],{"id":90611},"to-shell-and-back","To shell and back",[18,90614,90615],{},"As we build up our project, you will likely want to jump back and forth between your graphic user interface (like macOS Finder, Windows Explorer, etc.) and your shell program (like macOS Terminal, Windows PowerShell, etc). Here’s an easy guide for jumping between the two and always landing in the exact folder you need:",[7660,90617,90619],{"id":90618},"macos-finder-and-terminal","macOS: Finder and Terminal",[18,90621,90622],{},"To jump from Finder to Terminal while remaining in the same folder: 1. Within Finder, navigate to your intended folder. 2. Right-click inside the folder (or on the folder itself) to open a context menu. 3. Choose the “New Terminal at Folder” option.",[18,90624,90625,90626,90629],{},"(If you don’t see this option, go to your System Settings → Privacy & Security → Extensions → Finder → enable Terminal.) To jump from Terminal to Finder while remaining in the same folder: 1. Within Terminal, navigate to your intended folder. 2. Enter the following and press Enter. (Yes, include the ",[1031,90627,90628],{},"empty space"," followed by a period. In this context the period is an alias for “here”, as in “open here.”)",[498,90631,90633],{"className":86859,"code":90632,"language":86861,"meta":104,"style":104},"open .\n",[504,90634,90635],{"__ignoreMap":104},[507,90636,90637,90639],{"class":509,"line":510},[507,90638,119],{"class":576},[507,90640,90641],{"class":730}," .\n",[7660,90643,90645],{"id":90644},"linux-file-manager-and-terminal","Linux: File Manager and Terminal",[18,90647,90648],{},"To jump from File Manager to Terminal while remaining in the same folder (Ubuntu, Fedora, Debian, Arch, Mint, etc.): 1. Within File Manager, navigate to your intended folder. 2. Right-click inside the folder to open a context menu. 3. Choose the “Open in Terminal” option.",[18,90650,90651,90652,90629],{},"To jump from Terminal to File Manager while remaining in the same folder: 1. Within Terminal, navigate to your intended folder. 2. Enter the following and press Enter. (Yes, include the ",[1031,90653,90628],{},[498,90655,90657],{"className":86859,"code":90656,"language":86861,"meta":104,"style":104},"xdg-open .\n",[504,90658,90659],{"__ignoreMap":104},[507,90660,90661,90664],{"class":509,"line":510},[507,90662,90663],{"class":576},"xdg-open",[507,90665,90641],{"class":730},[7660,90667,90669],{"id":90668},"windows-explorer-and-powershell","Windows: Explorer and PowerShell",[18,90671,90672,90673,90676],{},"To jump from File Explorer to PowerShell while remaining in the same folder: 1. Within File Explorer, navigate to your intended folder. 2. Click the address bar. 3. Type ",[504,90674,90675],{},"powershell"," and press Enter.",[18,90678,90679],{},"To jump from PowerShell to Explorer while remaining in the same folder:",[27370,90681,90682,90685],{},[45,90683,90684],{},"Within PowerShell, navigate to your intended folder.",[45,90686,90687,90688,90690],{},"Enter the following and press Enter. (Yes, include the ",[1031,90689,90628],{}," followed by a period. In this context the period is an alias for “here”, as in “explore here.”)",[498,90692,90694],{"className":86859,"code":90693,"language":86861,"meta":104,"style":104},"explorer .\n",[504,90695,90696],{"__ignoreMap":104},[507,90697,90698,90701],{"class":509,"line":510},[507,90699,90700],{"class":576},"explorer",[507,90702,90641],{"class":730},[2513,90704,90706],{"id":90705},"initialize-the-project-with-uv","Initialize the project with UV",[18,90708,90709,90710,90712],{},"Now that we have a ",[504,90711,85760],{}," project folder on our Desktop (and know how to jump between our graphic interface and a shell), open a shell to your project’s folder and enter the following command:",[498,90714,90716],{"className":86859,"code":90715,"language":86861,"meta":104,"style":104},"uv init\n",[504,90717,90718],{"__ignoreMap":104},[507,90719,90720,90722],{"class":509,"line":510},[507,90721,86898],{"class":576},[507,90723,90724],{"class":730}," init\n",[18,90726,90727],{},"This little command packs quite a punch. It creates several files for us, some of which are hidden from view (in order to reduce clutter). Let’s have a look at the visible files first:",[42,90729,90730,90735,90740,90750],{},[45,90731,90732,90734],{},[154,90733,90595],{},". This file specifies what version of Python our project ought to use, as well as any depencies we decide to include later. This a human-editable file, it’s yours to update.",[45,90736,90737,90739],{},[154,90738,88565],{},". A minimal (yet executable) Python file that we can begin editing and build from. (We’ll run this file in just a moment!)",[45,90741,90742,90745,90746,90749],{},[154,90743,90744],{},"README.md",". An empty “Read me” ",[49,90747,70484],{"href":90748},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMarkdown"," file for documenting our project.",[45,90751,90752,90755,90756,90758,90759,90761,90762,90764],{},[154,90753,90754],{},"uv.lock",". In contrast to ",[504,90757,90595],{},", this file is machine-generated and should not be manually edited. While ",[504,90760,90595],{}," decribes our intent, ",[504,90763,90754],{}," is a detailed documentation of what packages and versions are actually in use.",[18,90766,90767],{},"The hidden files are also informative:",[42,90769,90770,90781,90791],{},[45,90771,90772,90775,90776,90780],{},[154,90773,90774],{},".python-version",". ",[49,90777,90779],{"href":90778},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FDoes_exactly_what_it_says_on_the_tin","Just what it says on the tin",": A file specifying the version of Python required.",[45,90782,90783,90785,90786,90790],{},[154,90784,87683],{},". As part of our project’s initialization, UV automatically created a local ",[49,90787,90789],{"href":90788},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FGit","git code repository"," for us, and included this handy list of files and file types to ignore in our commits.",[45,90792,90793,90796],{},[154,90794,90795],{},".venv",". An entire folder dedicated to specifying and maintaining our project’s virtual environment. Leave this folder and its content alone.",[2513,90798,90800],{"id":90799},"run-our-uv-project","Run our UV project",[18,90802,90803],{},"As UV was kind enough to generate a skeletal Python file for us, let’s take it for a test drive. Be sure that your shell is still within our project’s folder and enter the following:",[498,90805,90806],{"className":86859,"code":88569,"language":86861,"meta":104,"style":104},[504,90807,90808],{"__ignoreMap":104},[507,90809,90810,90812,90814,90816],{"class":509,"line":510},[507,90811,86898],{"class":576},[507,90813,86901],{"class":730},[507,90815,86904],{"class":730},[507,90817,88582],{"class":730},[18,90819,90820],{},"Depending on what you named your project folder, your shell should respond with something similar to:",[498,90822,90824],{"className":86859,"code":90823,"language":86861,"meta":104,"style":104},"Hello from qollab!\n",[504,90825,90826],{"__ignoreMap":104},[507,90827,90828,90831,90834],{"class":509,"line":510},[507,90829,90830],{"class":576},"Hello",[507,90832,90833],{"class":730}," from",[507,90835,90836],{"class":730}," qollab!\n",[18,90838,90839],{},[49,90840,90842],{"href":90841},"https:\u002F\u002Fen.wiktionary.org\u002Fwiki\u002Fcooking_with_gas","Now we’re cooking with gas!",[13,90844,90846],{"id":90845},"_5-pick-a-text-editor","5. Pick a text editor",[18,90848,90849,90850,90853,90854,947,90858,90862,90863,90867,90868,90871,90872,90875,90876,90880,90881,90885],{},"It’s time to start writing your own Python code, and that means editing text files. Python code is just ",[1031,90851,90852],{},"plain text",", after all. That means your operating system’s built-in apps (like ",[49,90855,90857],{"href":90856},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FTextEdit","Text Edit",[49,90859,90861],{"href":90860},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWindows_Notepad","Notepad",") are already enough to write and edit Python (provided you have ",[49,90864,90866],{"href":90865},"https:\u002F\u002Fsites.radford.edu\u002F~rstepno\u002F326\u002Ftextedit\u002Findex.html","rich text turned off",", of course). But a ",[1031,90869,90870],{},"robust"," code editing environment can actually make coding ",[1031,90873,90874],{},"enjoyable"," through modern conveniences like ",[49,90877,90879],{"href":90878},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSyntax_highlighting","syntax highlighting",", auto-",[49,90882,90884],{"href":90883},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FIndentation_style","indentation",", and more.",[18,90887,90888,90889,90893,90894,90898],{},"If you don’t already have a favorite text editor or ",[49,90890,90892],{"href":90891},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FIntegrated_development_environment","Integrated Development Environment (IDE)",", now’s the time to discover one that’s right for you. While Wikipedia provides a ",[49,90895,90897],{"href":90896},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FComparison_of_text_editors#Programming_features","comprehensive comparison of text editors",", this more curated list of free, cross-platform coding apps will help you get started.",[41852,90900,90901,90914],{},[41855,90902,90903],{},[41858,90904,90905,90908,90911],{},[41861,90906,90907],{},"Name",[41861,90909,90910],{"align":87757},"Friendliness",[41861,90912,90913],{},"Description",[41868,90915,90916,90930,90943,90956,90969,90983],{},[41858,90917,90918,90924,90927],{},[41873,90919,90920],{},[49,90921,90923],{"href":90922},"https:\u002F\u002Fcode.visualstudio.com","VS Code",[41873,90925,90926],{"align":87757},"✅",[41873,90928,90929],{},"Extension-driven code editor that balances approachability with serious IDE-level power.",[41858,90931,90932,90938,90940],{},[41873,90933,90934],{},[49,90935,90937],{"href":90936},"https:\u002F\u002Fwww.sublimetext.com","Sublime",[41873,90939,90926],{"align":87757},[41873,90941,90942],{},"Blazing-fast, minimalist editor famous for multi-cursor editing and near-instant responsiveness.",[41858,90944,90945,90951,90953],{},[41873,90946,90947],{},[49,90948,90950],{"href":90949},"https:\u002F\u002Fkate-editor.org","Kate",[41873,90952,90926],{"align":87757},[41873,90954,90955],{},"Capable, lightweight KDE editor with strong syntax highlighting and project features without IDE heaviness.",[41858,90957,90958,90964,90966],{},[41873,90959,90960],{},[49,90961,90963],{"href":90962},"https:\u002F\u002Fwww.geany.org","Geany",[41873,90965,90926],{"align":87757},[41873,90967,90968],{},"Small, simple IDE-style editor that offers compilation and tooling with minimal resource usage.",[41858,90970,90971,90977,90980],{},[41873,90972,90973],{},[49,90974,90976],{"href":90975},"https:\u002F\u002Fwww.gnu.org\u002Fsoftware\u002Femacs","Emacs",[41873,90978,90979],{"align":87757},"😅",[41873,90981,90982],{},"Deeply extensible, keyboard-centric editor that doubles as a programmable computing environment.",[41858,90984,90985,90991,90993],{},[41873,90986,90987],{},[49,90988,90990],{"href":90989},"https:\u002F\u002Fwww.vim.org","Vim",[41873,90992,90979],{"align":87757},[41873,90994,90995],{},"Modal, terminal-native editor optimized for extreme speed and precision once its commands are mastered.",[13,90997,88355],{"id":88354},[18,90999,91000],{},"You can open a shell. You can bounce between your shell and GUI while staying within your project’s code folder. You’ve installed UV and Python. You’ve created a UV Python project and run some Python code. With a good text editor in hand, you’re ready to start making some really productive mistakes, an exciting start to your journey. You’re ready to venture on to more Qollab tutorials:",[42,91002,91003,91008,91013],{},[45,91004,91005,53],{},[49,91006,88459],{"href":91007},"\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-basics",[45,91009,91010,53],{},[49,91011,91012],{"href":87180},"Setup the Qiskit SDK",[45,91014,91015,53],{},[49,91016,87150],{"href":86415},[953,91018,91019],{},"html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":104,"searchDepth":105,"depth":105,"links":91021},[91022,91023,91028,91033,91038,91043,91044],{"id":90159,"depth":105,"text":90160},{"id":90184,"depth":105,"text":90185,"children":91024},[91025,91026,91027],{"id":90205,"depth":540,"text":90206},{"id":90268,"depth":540,"text":90269},{"id":90306,"depth":540,"text":90307},{"id":90333,"depth":105,"text":90334,"children":91029},[91030,91031,91032],{"id":90371,"depth":540,"text":90372},{"id":90417,"depth":540,"text":90418},{"id":90428,"depth":540,"text":90429},{"id":90482,"depth":105,"text":90483,"children":91034},[91035,91036,91037],{"id":90503,"depth":540,"text":90504},{"id":90513,"depth":540,"text":90514},{"id":90542,"depth":540,"text":90543},{"id":90580,"depth":105,"text":90581,"children":91039},[91040,91041,91042],{"id":90611,"depth":540,"text":90612},{"id":90705,"depth":540,"text":90706},{"id":90799,"depth":540,"text":90800},{"id":90845,"depth":105,"text":90846},{"id":88354,"depth":105,"text":88355},[112,85993,85994,88481],[],{"username":86363,"name":87130,"role":87131,"avatar":104},{"image":91049,"alt":88481},"\u002F_content\u002Fimages\u002Fpython-setup\u002Fhero.webp",{},{"slug":90107,"title":91052,"desc":91053},"2 · Learn Python language basics","Python has become the de facto programming language of quantum computing. This guide highlights some basic Python commands and logic; foundational…","\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-setup","12 min read",[],{"title":91058,"description":88428},"Setup Python on your machine · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fpython-setup",[],"HKSfY9GAjazusWgHYpNKQ9dMAa0VjOLFsMA_Ngfv_FU",{"id":91063,"title":91064,"authors":91065,"body":91066,"breadcrumb":91626,"builders":91627,"byline":91628,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":90106,"description":90106,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":91629,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":590,"meta":91631,"navigation":133,"newsItems":116,"next":91632,"ogImage":116,"order":540,"outcomes":116,"path":90104,"publishDate":88450,"readingTime":993,"related":91635,"relatedProjects":116,"seo":91636,"stem":91638,"tags":91639,"track":86016,"trackName":85994,"__hash__":91640},"blog\u002Fblog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fqiskit-setup.md","Setup IBM’s Qiskit SDK",[86363],{"type":10,"value":91067,"toc":91616},[91068,91086,91090,91096,91100,91103,91116,91149,91153,91156,91182,91186,91204,91216,91222,91226,91229,91241,91244,91256,91260,91263,91276,91279,91297,91308,91322,91325,91343,91347,91353,91532,91558,91564,91577,91580,91603,91606,91610,91613],[18,91069,91070,91071,91075,91076,91078,91079,91082,91083,53],{},"IBM’s Qiskit is a leading quantum Software Development Kit (SDK) used as a foundation, not just for use with IBM hardware, but for various other quantum computing hardware architectures as well. IBM already has excellent ",[49,91072,91074],{"href":91073},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fen\u002Ftutorials","Qiskit tutorials"," available online for free. Our Qollab Qiskit guides take a specific, ",[49,91077,87272],{"href":87271},"-based approach to managing Python projects. If you have not installed UV (or Python), take a look at our previous guide: ",[49,91080,91081],{"href":86410},"Setup Python (and UV) on your machine",". For a crash course in Python fundamentals, see our guide: ",[49,91084,91085],{"href":91007},"Learn Python basics",[13,91087,91089],{"id":91088},"_1-create-a-new-project","1. Create a new project",[18,91091,91092,91093,90608],{},"Create a project folder on your Desktop titled ",[504,91094,91095],{},"our-qollab",[2513,91097,91099],{"id":91098},"initialize-our-app","Initialize our app",[18,91101,91102],{},"Open a shell to your project’s folder and enter the following command:",[498,91104,91106],{"className":86859,"code":91105,"language":86861,"meta":104,"style":104},"uv init --app\n",[504,91107,91108],{"__ignoreMap":104},[507,91109,91110,91112,91114],{"class":509,"line":510},[507,91111,86898],{"class":576},[507,91113,87320],{"class":730},[507,91115,87323],{"class":583},[18,91117,91118,91119,91122,91123,91126,91127,91130,91131,91134,91135,91138,91139,91141,91142,947,91145,91148],{},"This will initialize your new ",[49,91120,91121],{"href":87271},"UV-managed"," project. The ",[504,91124,91125],{},"uv init"," command has two main modes: 1. ",[154,91128,91129],{},"Library mode"," (default): For building a script that is primarily intended to be a reusable package, imported by other applications. 2. ",[154,91132,91133],{},"Application mode"," (using the ",[504,91136,91137],{},"--app"," flag): For building a script that is intended to be executable on its own. The ",[504,91140,91137],{}," flag instructs UV to assume the app will be run via ",[504,91143,91144],{},"uv run ...",[504,91146,91147],{},"uv run python -m ...",". It’s not meant to be installed as a dependency, and entry points matter more than exports.",[2513,91150,91152],{"id":91151},"select-a-python-version","Select a Python version",[18,91154,91155],{},"We want to be selective about which Python version our application uses because Qiskit and its various packages have particular compatibility requirements, and sometimes lag behind the most recent Python release. As of this writing, Python 3.12 is a safe choice for use. Enter the following into your shell to install Python 3.12 (if not already available) and explitly pin it as the version required by our application.",[498,91157,91159],{"className":86859,"code":91158,"language":86861,"meta":104,"style":104},"uv python install 3.12\nuv python pin 3.12\n",[504,91160,91161,91172],{"__ignoreMap":104},[507,91162,91163,91165,91167,91169],{"class":509,"line":510},[507,91164,86898],{"class":576},[507,91166,86904],{"class":730},[507,91168,87298],{"class":730},[507,91170,91171],{"class":583}," 3.12\n",[507,91173,91174,91176,91178,91180],{"class":509,"line":105},[507,91175,86898],{"class":576},[507,91177,86904],{"class":730},[507,91179,87311],{"class":730},[507,91181,91171],{"class":583},[7660,91183,91185],{"id":91184},"pitfalls-to-avoid","Pitfalls to avoid",[18,91187,91188,91189,86508,91193,91195,91196,91200,91201,91203],{},"Qiskit only supports ",[49,91190,91192],{"href":91191},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCPython","CPython",[1031,91194,90438],{}," ",[49,91197,91199],{"href":91198},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPyPy","PyPy",". You can confirm your Python versions via ",[504,91202,86898],{}," by entering the following into your shell:",[498,91205,91206],{"className":86859,"code":90520,"language":86861,"meta":104,"style":104},[504,91207,91208],{"__ignoreMap":104},[507,91209,91210,91212,91214],{"class":509,"line":510},[507,91211,86898],{"class":576},[507,91213,86904],{"class":730},[507,91215,90531],{"class":730},[18,91217,91218,91219,53],{},"If you’re upgrading across major Qiskit eras, don’t “upgrade in place” inside an old virtual environment. Instead, make a new virtual environment for each project using ",[504,91220,91221],{},"uv venv",[2513,91223,91225],{"id":91224},"create-a-virtual-environment","Create a virtual environment",[18,91227,91228],{},"Initializing a UV app will “lazily” create a virtual environment. That is, the virtual environment ought to create and activate on-demand when it is first needed. But out of precaution we’d like to manually do that now ourselves. Enter the following into your shell:",[498,91230,91232],{"className":86859,"code":91231,"language":86861,"meta":104,"style":104},"uv venv\n",[504,91233,91234],{"__ignoreMap":104},[507,91235,91236,91238],{"class":509,"line":510},[507,91237,86898],{"class":576},[507,91239,91240],{"class":730}," venv\n",[18,91242,91243],{},"At this point our shell may ask us to enter something similar the following in order to activate our virtual environment:",[498,91245,91247],{"className":86859,"code":91246,"language":86861,"meta":104,"style":104},"source .venv\u002Fbin\u002Factivate\n",[504,91248,91249],{"__ignoreMap":104},[507,91250,91251,91253],{"class":509,"line":510},[507,91252,86878],{"class":572},[507,91254,91255],{"class":730}," .venv\u002Fbin\u002Factivate\n",[13,91257,91259],{"id":91258},"_2-add-qiskit-packages","2. Add Qiskit packages",[18,91261,91262],{},"Enter the following into your shell to install Qiskit’s core package:",[498,91264,91266],{"className":86859,"code":91265,"language":86861,"meta":104,"style":104},"uv add qiskit\n",[504,91267,91268],{"__ignoreMap":104},[507,91269,91270,91272,91274],{"class":509,"line":510},[507,91271,86898],{"class":576},[507,91273,87354],{"class":730},[507,91275,77929],{"class":730},[18,91277,91278],{},"Let’s confirm that Qiskit has installed correctly. Enter the following into your shell and it should respond with a Qiskit version number:",[498,91280,91282],{"className":86859,"code":91281,"language":86861,"meta":104,"style":104},"uv run python -c \"import qiskit; print('Qiskit', qiskit.__version__)\"\n",[504,91283,91284],{"__ignoreMap":104},[507,91285,91286,91288,91290,91292,91294],{"class":509,"line":510},[507,91287,86898],{"class":576},[507,91289,86901],{"class":730},[507,91291,86904],{"class":730},[507,91293,87379],{"class":583},[507,91295,91296],{"class":730}," \"import qiskit; print('Qiskit', qiskit.__version__)\"\n",[18,91298,91299,91300,91302,91303,91307],{},"Our first Qiskit app is going to use a quantum ",[1031,91301,21959],{}," that runs on your local machine, as opposed to using a cloud-based simulator or actual quantum hardware. (Don’t worry, we’ll get to real quantum hardware soon enough.) Install Qiskit’s fast local simulator (",[49,91304,91306],{"href":91305},"https:\u002F\u002Fqiskit.github.io\u002Fqiskit-aer\u002F","Qiskit Aer",") by entering the following into your shell:",[498,91309,91311],{"className":86859,"code":91310,"language":86861,"meta":104,"style":104},"uv add qiskit-aer\n",[504,91312,91313],{"__ignoreMap":104},[507,91314,91315,91317,91319],{"class":509,"line":510},[507,91316,86898],{"class":576},[507,91318,87354],{"class":730},[507,91320,91321],{"class":730}," qiskit-aer\n",[18,91323,91324],{},"We can run a similiar sanity check for Aer as well:",[498,91326,91328],{"className":86859,"code":91327,"language":86861,"meta":104,"style":104},"uv run python -c \"import qiskit_aer; print('Qiskit Aer', qiskit_aer.__version__)\"\n",[504,91329,91330],{"__ignoreMap":104},[507,91331,91332,91334,91336,91338,91340],{"class":509,"line":510},[507,91333,86898],{"class":576},[507,91335,86901],{"class":730},[507,91337,86904],{"class":730},[507,91339,87379],{"class":583},[507,91341,91342],{"class":730}," \"import qiskit_aer; print('Qiskit Aer', qiskit_aer.__version__)\"\n",[13,91344,91346],{"id":91345},"_3-run-a-local-simulation","3. Run a local simulation",[18,91348,91349,91350,91352],{},"Now that we’ve installed some Qiskit packages, let’s put them to use. Open up your project’s ",[504,91351,88565],{}," Python script with your favorite text editor. Replace its contents with the following, and save it to disk:",[498,91354,91356],{"className":500,"code":91355,"language":502,"meta":104,"style":104},"from qiskit import QuantumCircuit\nfrom qiskit_aer import AerSimulator\n\ndef main():\n    qc = QuantumCircuit(2)\n    qc.h(0)\n    qc.cx(0, 1)\n    qc.measure_all()\n\n    sim = AerSimulator()\n    result = sim.run(qc, shots=1000).result()\n    counts = result.get_counts()\n    print(\"Counts:\", counts)\n\nif __name__ == \"__main__\":\n    main()\n",[504,91357,91358,91368,91380,91384,91392,91406,91418,91434,91442,91446,91458,91483,91495,91507,91511,91525],{"__ignoreMap":104},[507,91359,91360,91362,91364,91366],{"class":509,"line":510},[507,91361,529],{"class":513},[507,91363,532],{"class":517},[507,91365,514],{"class":513},[507,91367,537],{"class":517},[507,91369,91370,91372,91375,91377],{"class":509,"line":105},[507,91371,529],{"class":513},[507,91373,91374],{"class":517}," qiskit_aer ",[507,91376,514],{"class":513},[507,91378,91379],{"class":517}," AerSimulator\n",[507,91381,91382],{"class":509,"line":540},[507,91383,556],{"emptyLinePlaceholder":133},[507,91385,91386,91388,91390],{"class":509,"line":553},[507,91387,1370],{"class":513},[507,91389,73467],{"class":576},[507,91391,1930],{"class":517},[507,91393,91394,91396,91398,91400,91402,91404],{"class":509,"line":559},[507,91395,72833],{"class":517},[507,91397,573],{"class":572},[507,91399,577],{"class":576},[507,91401,580],{"class":517},[507,91403,584],{"class":583},[507,91405,587],{"class":517},[507,91407,91408,91410,91412,91414,91416],{"class":509,"line":566},[507,91409,21867],{"class":517},[507,91411,596],{"class":576},[507,91413,580],{"class":517},[507,91415,601],{"class":583},[507,91417,587],{"class":517},[507,91419,91420,91422,91424,91426,91428,91430,91432],{"class":509,"line":590},[507,91421,21867],{"class":517},[507,91423,615],{"class":576},[507,91425,580],{"class":517},[507,91427,601],{"class":583},[507,91429,622],{"class":517},[507,91431,625],{"class":583},[507,91433,587],{"class":517},[507,91435,91436,91438,91440],{"class":509,"line":610},[507,91437,21867],{"class":517},[507,91439,86659],{"class":576},[507,91441,781],{"class":517},[507,91443,91444],{"class":509,"line":634},[507,91445,556],{"emptyLinePlaceholder":133},[507,91447,91448,91451,91453,91456],{"class":509,"line":661},[507,91449,91450],{"class":517},"    sim ",[507,91452,573],{"class":572},[507,91454,91455],{"class":576}," AerSimulator",[507,91457,781],{"class":517},[507,91459,91460,91462,91464,91467,91469,91471,91473,91475,91477,91479,91481],{"class":509,"line":678},[507,91461,59864],{"class":517},[507,91463,573],{"class":572},[507,91465,91466],{"class":517}," sim.",[507,91468,22501],{"class":576},[507,91470,79863],{"class":517},[507,91472,68762],{"class":2155},[507,91474,573],{"class":572},[507,91476,79870],{"class":583},[507,91478,14176],{"class":517},[507,91480,23996],{"class":576},[507,91482,781],{"class":517},[507,91484,91485,91487,91489,91491,91493],{"class":509,"line":683},[507,91486,73551],{"class":517},[507,91488,573],{"class":572},[507,91490,84994],{"class":517},[507,91492,73558],{"class":576},[507,91494,781],{"class":517},[507,91496,91497,91499,91501,91504],{"class":509,"line":697},[507,91498,2060],{"class":572},[507,91500,580],{"class":517},[507,91502,91503],{"class":730},"\"Counts:\"",[507,91505,91506],{"class":517},", counts)\n",[507,91508,91509],{"class":509,"line":710},[507,91510,556],{"emptyLinePlaceholder":133},[507,91512,91513,91515,91518,91520,91523],{"class":509,"line":715},[507,91514,1645],{"class":513},[507,91516,91517],{"class":58306}," __name__",[507,91519,21255],{"class":572},[507,91521,91522],{"class":730}," \"__main__\"",[507,91524,1728],{"class":517},[507,91526,91527,91530],{"class":509,"line":721},[507,91528,91529],{"class":576},"    main",[507,91531,781],{"class":517},[18,91533,91534,91535,91538,91539,88086,91541,91543,91544,88102,91546,88105,91548,88108,91550,88113,91552,91554,91555,91557],{},"This Python script imports Qiskit and Qiskit’s “Aer” local simulator. The command ",[504,91536,91537],{},"QuantumCircuit(2)"," creates a quantum circuit composed of two qubit registers, both initialized to a | 0 ⟩ (“ket zero”) state. It then places a ",[49,91540,88085],{"href":88084},[504,91542,601],{},", flipping that qubit into superposition. Next, it places a ",[49,91545,88101],{"href":88100},[504,91547,601],{},[504,91549,625],{},[49,91551,88112],{"href":88111},[504,91553,601],{},"’s superposition across the two qubits, entangling them. Finally, we measure both qubit registers, collapsing the distributed superposition into a definitive value. The two possible measured values are | 00 ⟩ and | 11 ⟩ , each with a 50% probabilty of occurrence. This simulation is run 1,000 times (",[504,91556,88131],{},") and the results of these “shots” and printed to the command line.",[18,91559,91560,91561,91563],{},"Execute this new ",[504,91562,88565],{}," quantum script by entering the following into your shell:",[498,91565,91567],{"className":86859,"code":91566,"language":86861,"meta":104,"style":104},"uv run main.py\n",[504,91568,91569],{"__ignoreMap":104},[507,91570,91571,91573,91575],{"class":509,"line":510},[507,91572,86898],{"class":576},[507,91574,86901],{"class":730},[507,91576,88582],{"class":730},[18,91578,91579],{},"It will respond with output similar to the following:",[498,91581,91583],{"className":86859,"code":91582,"language":86861,"meta":104,"style":104},"Counts: {'00': 517, '11': 483}\n",[504,91584,91585],{"__ignoreMap":104},[507,91586,91587,91590,91593,91596,91598,91601],{"class":509,"line":510},[507,91588,91589],{"class":576},"Counts:",[507,91591,91592],{"class":730}," {'00':",[507,91594,91595],{"class":730}," 517,",[507,91597,88250],{"class":730},[507,91599,91600],{"class":583}," 483",[507,91602,23875],{"class":730},[18,91604,91605],{},"Congratulations. You’ve just executed an incredibly convoluted coin-flip routine!",[13,91607,91609],{"id":91608},"coming-soon","Coming soon",[18,91611,91612],{},"Stay tuned for our guide to running quantum simulations in the cloud, performing operations on actual quantum hardware, and making use of IonQ hardware architectures.",[953,91614,91615],{},"html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}",{"title":104,"searchDepth":105,"depth":105,"links":91617},[91618,91623,91624,91625],{"id":91088,"depth":105,"text":91089,"children":91619},[91620,91621,91622],{"id":91098,"depth":540,"text":91099},{"id":91151,"depth":540,"text":91152},{"id":91224,"depth":540,"text":91225},{"id":91258,"depth":105,"text":91259},{"id":91345,"depth":105,"text":91346},{"id":91608,"depth":105,"text":91609},[112,85993,85994,91064],[],{"username":86363,"name":87130,"role":87131,"avatar":104},{"image":91630,"alt":91064},"\u002F_content\u002Fimages\u002Fqiskit-setup\u002Fhero.webp",{},{"slug":88412,"title":91633,"desc":91634},"4 · Setup and simulate with IonQ","Create and leverage a free IonQ account to run an example quantum circuit on IonQ’s cloud simulator. Builds upon our previous tutorials for…",[],{"title":91637,"description":90106},"Setup IBM’s Qiskit SDK · Building your first Qollab project","blog\u002Flearn\u002Fbuilding-your-first-qollab-project\u002Fqiskit-setup",[],"jUMXsmGEDsv9ZSFtclHqtYskcpHnf-c1E6vnLprtyE4",{"id":91642,"title":91643,"authors":91644,"body":91645,"breadcrumb":91652,"builders":91653,"byline":91654,"challenge":116,"courseAuthor":91655,"courseLead":91658,"dek":91659,"description":91660,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":116,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":88442,"lessonCount":116,"meta":91661,"navigation":133,"newsItems":116,"next":116,"ogImage":116,"order":116,"outcomes":91662,"path":91667,"publishDate":91668,"readingTime":116,"related":91669,"relatedProjects":116,"seo":91670,"stem":91672,"tags":91673,"track":91674,"trackName":116,"__hash__":91675},"blog\u002Fblog\u002Flearn\u002Fquantum-foundations.md","Quantum foundations",[86363],{"type":10,"value":91646,"toc":91650},[91647],[18,91648,91649],{},"Quantum computing is easier than you might think. (Remember it’s just computing, it’s not quantum physics!) Some quick math brush-ups and few concept primers…",{"title":104,"searchDepth":105,"depth":105,"links":91651},[],[112,85993,91643],[],{"username":86363,"name":87130,"role":104,"avatar":104},{"name":87130,"role":104,"bio":88435,"avatar":104,"links":91656},[91657],{"label":73068,"href":88438},"Start from zero and build up the math that quantum computing actually runs on. By the end you can read qubits, gates, and circuits well enough to open the Code Playground and run your first circuit on real hardware.","Qubits, gates, and what actually makes quantum different. No prior quantum needed.","A free five-lesson course on the foundations of quantum computing: qubits, gates, matrices, and complex numbers. About an hour, no account needed.",{},[91663,91664,91665,91666],"Why quantum computers matter, and what they are actually for","The math they run on: complex numbers, matrices, and vectors","What a qubit really is: state vectors, superposition, the Bloch sphere","The core quantum gates and how they transform qubits","\u002Fblog\u002Flearn\u002Fquantum-foundations","2025-12-08",[],{"title":91671,"description":91660},"Quantum foundations · Learn by building","blog\u002Flearn\u002Fquantum-foundations",[],"quantum-foundations","RrMKnpvmNKTVZa9KQK7EOU8hXvJvKpo3cJkBPfhGkEM",{"id":91677,"title":91678,"authors":91679,"body":91680,"breadcrumb":91768,"builders":91769,"byline":91770,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":91771,"description":91771,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":91772,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":559,"meta":91774,"navigation":133,"newsItems":116,"next":91775,"ogImage":116,"order":105,"outcomes":116,"path":91779,"publishDate":91668,"readingTime":72277,"related":91780,"relatedProjects":116,"seo":91781,"stem":91783,"tags":91784,"track":91674,"trackName":91643,"__hash__":91785},"blog\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers.md","Complex numbers",[86363],{"type":10,"value":91681,"toc":91763},[91682,91685,91689,91697,91702,91705,91712,91716,91726,91733,91738,91741,91746,91750,91758],[18,91683,91684],{},"We’ll start with what you know: regular numbers. Then we’ll introduce “imaginary” numbers. A complex number is just a combination of a regular number with an imaginary one. Let’s go.",[13,91686,91688],{"id":91687},"real-numbers-ℝ","Real numbers ( ℝ )",[18,91690,91691,91692,91696],{},"Our regular, ordinary, everyday numbers are called ",[49,91693,91695],{"href":91694},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FReal_number","real numbers",". These include integers and decimals. You can visualize real numbers as existing along an infinite number line, with zero in the middle, positive numbers counting up forever to infinity on the right, and negative numbers doing the exact opposite on the left.",[831,91698],{"alt":91699,"caption":91700,"no":104,"src":91701},"A horizontal number line of Real Numbers with values labeled from -5 (left) to +5 (right) and arrows on either extreme indicating that these numbers extend infinitely in either direction.","The Real number line.","\u002F_content\u002Fimages\u002Fcomplex-numbers\u002Freal-numberline-3.webp",[18,91703,91704],{},"When a real number is multiplied by itself the product is always positive. For example, if we choose the number 2 we see that 2 × 2 = 4. Similarly, had we chosen the negative number -2, the product would still be positive because two negative numbers multiplied together also produce a positive result; -2 × -2 = 4. For brevity we could rewrite these equations as 22 = 4 and (-2)2 = 4, respectively.",[18,91706,54828,91707,91711],{},[49,91708,91710],{"href":91709},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FSquare_root","square root"," of a real number has two possible answers. The square root of 4, for example, is both 2 and -2 because both are solutions for x in the equation x = 4.",[13,91713,91715],{"id":91714},"imaginary-numbers-𝕀","Imaginary numbers ( 𝕀 )",[18,91717,91718,91719,91722,91723,91725],{},"But suppose we wanted to find the square root of a ",[1031,91720,91721],{},"negative"," number. Is there any number that could solve for x in the equation x = -4? Sadly, there is not. Or more precisely: there is not any ",[1031,91724,59914],{}," solution for the square root of a negative number.",[18,91727,91728,91732],{},[49,91729,91731],{"href":91730},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FImaginary_number","Imaginary numbers"," might be considered an “intermediate impossible.” The symbol i is defined as the imaginary solution to the equation x = -1, therefore i2 = -1. With this imaginary device we now have a solution to the above equation x = -4 and that solution is 2i. (And also -2i, of course. We can indicate this “plus or minus” possibility as ±2i.) Let’s inspect this more closely.",[87238,91734,91735],{},[18,91736,91737],{},"x = -4 x = 4 × -1 x = 4 × -1 x = ±2 × -1 x = ±2 × i x = ±2i",[18,91739,91740],{},"2i is an imaginary number that consists of a real number multiplier, 2, and our imaginary solution to -1, called i. Like real numbers, imaginary numbers also exist along an infinite number line. We plotted our real number line horizontally, so let’s plot our imaginary number line vertically.",[831,91742],{"alt":91743,"caption":91744,"no":104,"src":91745},"A vertical number line of Imaginary Numbers with values labeled from -5i (bottom) to +5 (top) and arrows on either extreme indicating that these numbers extend infinitely in either direction.","The Imaginary number line.","\u002F_content\u002Fimages\u002Fcomplex-numbers\u002Fimaginary-numberline-3.webp",[13,91747,91749],{"id":91748},"complex-numbers-ℂ","Complex numbers ( ℂ )",[18,91751,91752,91753,91757],{},"We just saw that multiplying a real number by i yields an imaginary number. But what if you add a real number to an imaginary one? Things get complex. A ",[49,91754,91756],{"href":91755},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FComplex_number","complex number"," is a number that can be expressed in the form a + bi, where a is the real component and bi is the imaginary component. Some examples might be 1 + 2i or 3 - 4i.",[831,91759],{"alt":91760,"caption":91761,"no":104,"src":91762},"Complex plane diagram.","The Complex plane.","\u002F_content\u002Fimages\u002Fcomplex-numbers\u002Fcomplex-plane-3.webp",{"title":104,"searchDepth":105,"depth":105,"links":91764},[91765,91766,91767],{"id":91687,"depth":105,"text":91688},{"id":91714,"depth":105,"text":91715},{"id":91748,"depth":105,"text":91749},[112,85993,91643,91678],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Complex numbers are a key part of orchestrating quantum algorithms, and you can learn what they are in just a few minutes.",{"image":91773,"alt":91678},"\u002F_content\u002Fimages\u002Fcomplex-numbers\u002Fhero.webp",{},{"slug":91776,"title":91777,"desc":91778},"\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fmatrices","3 · Matrices","Matrices are the mathematical building blocks for quantum bits, quantum gates, and quantum circuits.","\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers",[],{"title":91782,"description":91771},"Complex numbers · Quantum foundations","blog\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers",[],"p6kGszjbHdzS19ERzfnn4xvISFRCcC9_QjrXTuMwNZA",{"id":91787,"title":91788,"authors":91789,"body":91790,"breadcrumb":92196,"builders":92197,"byline":92198,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":92199,"description":92199,"draft":125,"extension":126,"eyebrow":116,"finish":92200,"fork":116,"hero":92201,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":559,"meta":92203,"navigation":133,"newsItems":116,"next":92204,"ogImage":116,"order":559,"outcomes":116,"path":92206,"publishDate":91668,"readingTime":92207,"related":92208,"relatedProjects":116,"seo":92209,"stem":92211,"tags":92212,"track":91674,"trackName":91643,"__hash__":92213},"blog\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fgates.md","Quantum logic gates",[86363],{"type":10,"value":91791,"toc":92178},[91792,91798,91802,91818,91839,91846,91867,91870,91907,91911,91915,91918,91921,91940,91944,91954,91958,91973,91976,91999,92003,92018,92021,92025,92036,92040,92044,92050,92054,92071,92075,92116,92120,92131,92135,92141,92145,92168,92172],[18,91793,91794,91795,53],{},"We’ll learn how quantum gates can act on a single qubit or multiple qubits in order to compute. While there’s no need to memorize the exact values that each gate uses to perform its operations, understanding the effect of some basic quantum gates is a solid foundation for understanding (and coding your own) ",[49,91796,86380],{"href":91797},"\u002Flearn\u002Fbuilding-your-first-qollab-project",[13,91799,91801],{"id":91800},"matrices-all-the-way-down","Matrices, all the way down",[18,91803,91804,91805,91807,91808,91812,91813,91817],{},"A quantum computer is just a collection of ",[49,91806,88068],{"href":88067},". These qubits hold probability values for resolving to either 0 or 1. A quantum computer is able to calculate things by changing the value of its qubits over time. We tell the computer exactly how it should change the value of a qubit by instructing it to “walk through” a series of ",[49,91809,91811],{"href":91810},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate","quantum gates",". As the qubit ",[49,91814,91816],{"href":91815},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FThe_Gates","“walks through” each gate"," its value is changed based on the type of gate it is passing through.",[18,91819,91820,91821,91824,91825,91829,91830,91833,91834,91838],{},"Contrary to what pop-science might tell you, there is nothing random or unpredictable about this process. Mathematically, a ",[49,91822,91823],{"href":88067},"qubit"," is just a ",[49,91826,91828],{"href":91827},"\u002Flearn\u002Fquantum-foundations\u002Fmatrices","matrix",". Similarly, a gate is also just a matrix. To apply a gate to a qubit is to ",[1031,91831,91832],{},"multiply"," these matrices together. To demonstrate this, let’s begin with a ",[49,91835,91837],{"href":91836},"\u002Flearn\u002Fquantum-foundations\u002Fqubits#named-couples","“Horizontal” qubit",", commonly thought of as representing “zero” or “off.” It has the following matrix form:",[18,91840,91841,91842,91845],{},"We would like to flip the value of this qubit from “off” to “on.” The result will be a ",[49,91843,91844],{"href":91836},"“Vertical” qubit"," with the following matrix form:",[18,91847,91848,91849,91852,91853,91856,91857,91861,91862,91866],{},"By looking at the pair of numbers that represent each qubit, you can see that a ",[49,91850,91851],{"href":91836},"Horizontal qubit"," is the inverse (or “flipped”) version of a ",[49,91854,91855],{"href":91836},"Vertical qubit",". In order to “flip” from one to the other we must apply a ",[49,91858,91860],{"href":91859},"#pauli-x-gate","Pauli X gate"," to our qubit. Because Pauli X gates have the effect of “flipping” the value of a qubit, they are often thought of as the quantum equivalent of a ",[49,91863,91865],{"href":91864},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FInverter_(logic_gate)","classical “NOT” gate",". Pauli X gates have the following matrix form:",[18,91868,91869],{},"We can now apply the Pauli X gate to the Horizontal qubit by multiplying their matrices together. It’s okay if your matrix multiplication is rusty, that’s what computers are for! The important part is just to recall that both qubits and quantum gates can be represented by matrices, and so can be multiplied together.",[18,91871,91872,91873,91875,91876,91879,91880,91883,91884,91195,91886,91890,91891,91893,91894,91896,91897,91893,91899,91901,91902,91906],{},"As anticipated, the resulting product matrix represents a ",[49,91874,91855],{"href":91836},". Note how the gate’s matrix is the ",[1031,91877,91878],{},"first"," factor (all the way to the left) and the qubit’s matrix is the ",[1031,91881,91882],{},"second"," factor (second in from the left, with the resulting product to the right of the equals sign). This order matters because matrix multiplication is ",[154,91885,90438],{},[49,91887,91889],{"href":91888},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCommutative_property#Commutative_operations_in_mathematics","commutative",". With plain numbers (that is, numbers not contained within matrices), the order of the factors does not change the product outcome. a × b = b × a But with matrices, a different order yields a different outcome. ",[507,91892,49],{}," × ",[507,91895,13102],{}," ≠ ",[507,91898,13102],{},[507,91900,49],{}," See ",[49,91903,91905],{"href":91904},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMatrix_multiplication","matrix multiplication"," for an in-depth explanation.",[13,91908,91910],{"id":91909},"single-qubit-gates","Single-qubit gates",[2513,91912,91914],{"id":91913},"indentity-gate","Indentity gate",[18,91916,91917],{},"An Identity gate has no effect on the value of the qubit it operates on; equivalent to multiplying a value by one. (Generally when a circuit is created from text or another source, any included identity gates are ignored. It is included here for completeness.)",[2513,91919,91860],{"id":91920},"pauli-x-gate",[18,91922,54828,91923,91926,91927,91931,91932,91934,91935,91939],{},[49,91924,91860],{"href":91925},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Pauli-X_gate"," represents a rotation on the ",[49,91928,91930],{"href":91929},"\u002Flearn\u002Fquantum-foundations\u002Fqubits#bloch-sphere","Bloch sphere"," around the ",[154,91933,7731],{},"-axis by π radians. It is the quantum equivalent of the ",[49,91936,91938],{"href":91937},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FInverter%3C\u002Fem%3E(logic_gate)","classical NOT gate"," in that it maps | 0 ⟩ to | 1 ⟩ and | 1 ⟩ to | 0 ⟩ .",[2513,91941,91943],{"id":91942},"pauli-y-gate","Pauli Y gate",[18,91945,54828,91946,91926,91949,91931,91951,91953],{},[49,91947,91943],{"href":91948},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Pauli-Y_gate",[49,91950,91930],{"href":91929},[154,91952,7746],{},"-axis by π radians. It maps | 0 ⟩ to i | 1 ⟩ and | 1 ⟩ to -i | 0 ⟩ .",[2513,91955,91957],{"id":91956},"pauli-z-gate","Pauli Z gate",[18,91959,54828,91960,91926,91963,91931,91965,91967,91968,91972],{},[49,91961,91957],{"href":91962},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Pauli-Z_gate",[49,91964,91930],{"href":91929},[154,91966,7764],{},"-axis by π radians. It is a special case of a ",[49,91969,91971],{"href":91970},"#phase-shift-gates","Phase shift gate"," where ϕ = π, and is therefore sometimes referred to as a “phase-flip” gate. It leaves the basis state | 0 ⟩ unchanged and maps | 1 ⟩ to - | 1 ⟩ .",[2513,91974,88085],{"id":91975},"hadamard-gate",[18,91977,91978,91979,91983,91984,91987,91988,91990,91991,91995,91996,14176],{},"Applies a ",[49,91980,91982],{"href":91981},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Hadamard_(H)_gate","Hadamard transform"," to a single qubit. For the basis qubit states of | 0 ⟩ and | 1 ⟩ this has the effect of putting a qubit into ",[49,91985,88093],{"href":91986},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_superposition",". It represents a rotation on the ",[49,91989,91930],{"href":91929}," around the Z-axis by π radians, followed by a rotation around the Y-axis by π ÷ 2 radians. This maps the basis state | 0 ⟩ to | 0 ⟩ + | 1 ⟩ 2 (also referred to as | + ⟩ or ",[49,91992,91994],{"href":91993},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBra%E2%80%93ket_notation","“ket plus”",") and | 1 ⟩ to | 0 ⟩ - | 1 ⟩ 2 (also referred to as | - ⟩ or ",[49,91997,91998],{"href":91993},"“ket minus”",[2513,92000,92002],{"id":92001},"phase-shift-gates","Phase shift gates",[18,92004,92005,92009,92010,92012,92013,91931,92015,92017],{},[49,92006,92008],{"href":92007},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Phase_shift_gates","Phase gates"," are a family of quantum gates that employ the variable ϕ (phi) to represent tracing a horizontal arc (a line of latitude) of ϕ radians around the ",[49,92011,91930],{"href":91929},". They leave the basis state | 0 ⟩ unchanged and map | 1 ⟩ to eiφ | 1 ⟩ . The probability of measuring a | 0 ⟩ or | 1 ⟩ is unchanged after applying a phase shift gate, however modifying the phase of a quantum state (thankfully) does have implications within a quantum algorithm. The example form shown here represents a rotation on the ",[49,92014,91930],{"href":91929},[154,92016,7764],{},"-axis of π ÷ 2 radians.",[18,92019,92020],{},"Remember that ϕ (phi) is a variable here. You can substitute whatever values you would like for ϕ without changing the gate’s properties as described above.",[2513,92022,92024],{"id":92023},"t-gate","T gate",[18,92026,92027,92028,92032,92033,92035],{},"The T gate is also known as the “π÷8” gate and is a special case of a Phase shift gate where the ϕ (phi) variable is set to π÷4. (But why is it called “π÷8” when it actually divides π by 4? ",[49,92029,92031],{"href":92030},"https:\u002F\u002Fwww.quora.com\u002FWhy-is-the-quantum-T-gate-called-pi-8-gate-as-it-only-adds-a-phase-difference-of-pi-4-instead-of-pi-8-to-the-state-vector-1","It’s a little mathy",".) Like all phase shift gates, it represents a rotation on the Bloch sphere around the ",[154,92034,7764],{},"-axis.",[13,92037,92039],{"id":92038},"multi-qubit-gates","Multi-qubit gates",[2513,92041,92043],{"id":92042},"swap-gate","Swap gate",[18,92045,54828,92046,92049],{},[49,92047,92043],{"href":92048},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Swap_(SWAP)_gate"," swaps the value of two qubits. It is defined here with respect to the bases | 00 ⟩ , | 01 ⟩ , | 10 ⟩ , and | 11 ⟩ .",[2513,92051,92053],{"id":92052},"squareroot-swap-gate","Squareroot swap gate",[18,92055,54828,92056,92060,92061,92064,92065,92067,92068,92070],{},[49,92057,92059],{"href":92058},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Square_root_of_Swap_gate_(%E2%88%9ASWAP)","√Swap gate"," performs ",[1031,92062,92063],{},"half"," of a swap between two qubits. It is ",[1031,92066,90438],{}," maximally entangling. More than one application of it is required to produce a ",[49,92069,88112],{"href":88111}," from its product states. As with the Swap gate, is defined here with respect to the bases | 00 ⟩ , | 01 ⟩ , | 10 ⟩ , and | 11 ⟩ .",[2513,92072,92074],{"id":92073},"controlled-gates","Controlled gates",[18,92076,92077,92080,92081,10799,92083,92085,92086,92089,92090,92094,92095,80807,92099,947,92102,92105,92106,92109,92110,92112,92113,92115],{},[49,92078,92074],{"href":92079},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_logic_gate#Controlled_gates"," act on two or more qubits, where one qubit acts as the “target” of the operation and the remaining qubits act as “controls” determining ",[1031,92082,1645],{},[1031,92084,86127],{}," that target qubit is operated upon. In its most elementary form (a ",[49,92087,88101],{"href":92088},"#controlled-not-gate","), a controlled gate operation acts as a sort of ",[49,92091,92093],{"href":92092},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FConditional_(computer_programming)","“if” statement",", operating on the target qubit only when that “if” statement is satisfied, or to the degree with which it is satisfied. (Because qubits represent probabilities, they are not limited to strict ",[49,92096,92098],{"href":92097},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBoolean_algebra","Boolean values",[504,92100,92101],{},"YES",[504,92103,92104],{},"NO",". You might imagine the “if” statement being ",[1031,92107,92108],{},"partially"," satisifed, and thus ",[1031,92111,92108],{}," operating on the target qubit.) Through this process, controlled gates have the ability to ",[49,92114,79688],{"href":88119}," and disentangle qubits.",[7660,92117,92119],{"id":92118},"controlled-not-gate","Controlled NOT gate",[18,92121,92122,92123,92127,92128,92130],{},"The foundational example of a controlled gate is the ",[49,92124,92126],{"href":92125},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FControlled_NOT_gate","Controlled-Not (CNOT) gate",". The CNOT gate accepts two qubits as input, a control qubit and a target qubit, only operating on the target qubit according to the state of the control qubit. If the control qubit’s state is | 0 ⟩ (“off”) the target qubit remains untouched. However, if the control cubit’s state is | 1 ⟩ (“on”) then the target qubit’s state will be inverted by a ",[49,92129,91860],{"href":91859},". Of course, a qubit’s state is not limited to | 0 ⟩ or | 1 ⟩ and therein lies the fun. Its matrix representation is akin to an identity matrix and an inversion matrix globbed together:",[7660,92132,92134],{"id":92133},"controlled-swap-fredkin-gate","Controlled Swap (Fredkin) gate",[18,92136,54828,92137,92140],{},[49,92138,92134],{"href":92139},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FFredkin_gate"," operates on three qubits, using one as a control and two as targets. True to its name, it swaps the states of the two target qubits according to the state of the control bit. As we can see here, more qubits means more matrix values.",[7660,92142,92144],{"id":92143},"toffolli-ccnot-gate","Toffolli (CCNOT) gate",[18,92146,54828,92147,92151,92152,92156,92157,92160,92161,92163,92164,92167],{},[49,92148,92150],{"href":92149},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FToffoli_gate","Toffolli gate"," is also known as the CCNOT gate, a “controlled-controlled-not” gate. Like the ",[49,92153,92155],{"href":92154},"#controlled-swap-fredkin-gate","Controlled-swap “Fredkin” gate",", it operates on 3 qubits. Here we use ",[1031,92158,92159],{},"two"," control qubits to apply a ",[49,92162,91860],{"href":91859}," to a ",[1031,92165,92166],{},"single"," target qubit.",[13,92169,92171],{"id":92170},"ungated","Ungated",[18,92173,92174,92175,53],{},"Now that you’ve had a crash course in quantum gates, it’s probably a good time to step away from the screen, take a walk, and let some of this sink in. When you’re ready when can begin to explore ",[49,92176,92177],{"href":91797},"coding quantum circuits",{"title":104,"searchDepth":105,"depth":105,"links":92179},[92180,92181,92190,92195],{"id":91800,"depth":105,"text":91801},{"id":91909,"depth":105,"text":91910,"children":92182},[92183,92184,92185,92186,92187,92188,92189],{"id":91913,"depth":540,"text":91914},{"id":91920,"depth":540,"text":91860},{"id":91942,"depth":540,"text":91943},{"id":91956,"depth":540,"text":91957},{"id":91975,"depth":540,"text":88085},{"id":92001,"depth":540,"text":92002},{"id":92023,"depth":540,"text":92024},{"id":92038,"depth":105,"text":92039,"children":92191},[92192,92193,92194],{"id":92042,"depth":540,"text":92043},{"id":92052,"depth":540,"text":92053},{"id":92073,"depth":540,"text":92074},{"id":92170,"depth":105,"text":92171},[112,85993,91643,91788],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Quantum logic gates are the means of “getting work done” on a quantum computer. They perform computation by altering the values of quantum bits (“qubits”).","That’s Quantum foundations, start to finish.",{"image":92202,"alt":91788},"\u002F_content\u002Fimages\u002Fgates\u002Fhero.webp",{},{"slug":91054,"title":85994,"desc":92205},"Set up Python and Qiskit, run your first circuit on real IonQ hardware, then publish your project to Qollab.","\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fgates","19 min read",[],{"title":92210,"description":92199},"Quantum logic gates · Quantum foundations","blog\u002Flearn\u002Fquantum-foundations\u002Fgates",[],"lj0dFuGB70GbmrrYMH9tGLvx_mN6vH9iiLvjJs5j3_w",{"id":92215,"title":92216,"authors":92217,"body":92218,"breadcrumb":92313,"builders":92314,"byline":92315,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":91778,"description":91778,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":92316,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":559,"meta":92318,"navigation":133,"newsItems":116,"next":92319,"ogImage":116,"order":540,"outcomes":116,"path":91776,"publishDate":91668,"readingTime":72277,"related":92323,"relatedProjects":116,"seo":92324,"stem":92326,"tags":92327,"track":91674,"trackName":91643,"__hash__":92328},"blog\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fmatrices.md","Matrices",[86363],{"type":10,"value":92219,"toc":92307},[92220,92227,92231,92237,92240,92244,92252,92256,92267,92286,92289],[18,92221,92222,92223,92226],{},"This quick review of matrices will prime you to learn what qubits actually represent, including a concrete, non-",[1031,92224,92225],{},"woo-woo"," definition of superposition. Let’s get started.",[13,92228,92230],{"id":92229},"grid-of-numbers","Grid of numbers",[18,92232,90592,92233,92236],{},[49,92234,91828],{"href":92235},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMatrix_(mathematics)"," is just a grid of numbers; rows and columns containing values. Matrices can be of any size. Here’s an example of a 3×2 matrix. It is 3 columns wide and 2 rows tall, containing the values 1 through 6.",[18,92238,92239],{},"When describing the dimensions of a matrix we always specify the number of rows first, then the number of columns. The above is an example of a 3×2 matrix, while below is a matrix containing similar data, but in a 2×3 configuration.",[13,92241,92243],{"id":92242},"order-matters","Order matters",[18,92245,92246,92247,92251],{},"For our purposes, we’ll express our matrices in ",[49,92248,92250],{"href":92249},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRow-_and_column-major_order","row-major order",". This means we read the values just as they are ordered above, starting with the top-most row, reading values from left to right, then proceeding to the next row down and repeating that process. Our choice of row-major order makes reading and writing matrix values more akin to reading and writing in English; easier to type and program here.",[13,92253,92255],{"id":92254},"vectors-are-slices","Vectors are slices",[18,92257,92258,92259,92262,92263,92266],{},"While a matrix is a two-dimensional ",[1031,92260,92261],{},"grid"," of numbers, a vector is more like a ",[1031,92264,92265],{},"slice"," of numbers, such as a “skinny” matrix that is only one column wide, or a “flat” matrix that is only one row high. Let’s look at some examples of matrices that are simultaneously vectors.",[18,92268,92269,92270,92273,92274,92277,92278,92281,92282,92285],{},"While the above 2×2 matrix is not a vector, you could say that it ",[1031,92271,92272],{},"contains"," vectors: Two ",[1031,92275,92276],{},"column"," vectors or two ",[1031,92279,92280],{},"row"," vectors. Because vectors are just a type of matrix we can add them, multiply them, and so on, just like any other matrix. Vectors will play a prominent role in defining ",[49,92283,92284],{"href":88067},"Qubits"," and expressing the state of a quantum circuit.",[13,92287,91678],{"id":92288},"complex-numbers",[18,92290,92291,92292,92296,92297,1403,92300,92302,92303,92306],{},"In addition to storing regular numbers, matrices can contain ",[49,92293,92295],{"href":92294},"\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers","complex numbers",". This is both useful and ",[1031,92298,92299],{},"necessary",[49,92301,92284],{"href":88067}," are really just a pair of complex numbers that we store in a 1×2 matrix. So, yes, the example matrices above are each ",[1031,92304,92305],{},"larger"," than a qubit!",{"title":104,"searchDepth":105,"depth":105,"links":92308},[92309,92310,92311,92312],{"id":92229,"depth":105,"text":92230},{"id":92242,"depth":105,"text":92243},{"id":92254,"depth":105,"text":92255},{"id":92288,"depth":105,"text":91678},[112,85993,91643,92216],[],{"username":86363,"name":87130,"role":87131,"avatar":104},{"image":92317,"alt":92216},"\u002F_content\u002Fimages\u002Fmatrices\u002Fhero.webp",{},{"slug":92320,"title":92321,"desc":92322},"\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fqubits","4 · Qubits (quantum bits)","Like bits in classical computing, qubits are the fundamental containers for storing value in a quantum circuit. These stored values can be altered…",[],{"title":92325,"description":91778},"Matrices · Quantum foundations","blog\u002Flearn\u002Fquantum-foundations\u002Fmatrices",[],"igq53qzMMNKclGgfZBiX1sYPhHgDco44qyZnX1Uss7Y",{"id":92330,"title":92331,"authors":92332,"body":92333,"breadcrumb":92798,"builders":92799,"byline":92800,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":92801,"description":92802,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":92803,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":559,"meta":92805,"navigation":133,"newsItems":116,"next":92806,"ogImage":116,"order":553,"outcomes":116,"path":92320,"publishDate":91668,"readingTime":92809,"related":92810,"relatedProjects":116,"seo":92811,"stem":92813,"tags":92814,"track":91674,"trackName":91643,"__hash__":92815},"blog\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fqubits.md","Qubits (quantum bits)",[86363],{"type":10,"value":92334,"toc":92788},[92335,92338,92342,92353,92384,92403,92407,92410,92418,92426,92430,92446,92454,92462,92468,92472,92481,92509,92512,92535,92554,92557,92561,92577,92592,92606,92613,92616,92639,92651,92668,92671,92674,92677,92680,92686,92690,92701,92746,92754,92769,92772,92784],[18,92336,92337],{},"We’ll learn that qubits are mathematically simple structures, yet provide tremendous computational power. While there are multiple ways to implement physical qubits, we’re only interested in the mathematical concept of a qubit. Regardless of what method a quantum computer uses to implement its physical qubits, how they operate as a computing tools remains the same.",[13,92339,92341],{"id":92340},"perfect-pairs","Perfect pairs",[18,92343,90592,92344,92347,92348,92352],{},[49,92345,91823],{"href":92346},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQubit"," is just a pair of numbers. That’s it, two ",[49,92349,92351],{"href":92350},"https:\u002F\u002Fyoutu.be\u002FYYOKMUTTDdA","shiny happy number values, holding hands",". Let’s call the first number “alpha” and the second number “beta.” Look at this handsome couple:",[18,92354,92355,92356,92358,92359,92362,92363,92365,92366,92370,92371,92373,92374,88073,92376,92379,92380,92383],{},"We can package alpha and beta together by storing them in a very small ",[49,92357,91828],{"href":91827}," that is only ",[154,92360,92361],{},"one"," unit wide and ",[154,92364,92159],{}," units tall, and because this matrix is only one unit wide it is not merely a matrix, it’s also a ",[49,92367,92369],{"href":92368},"\u002Flearn\u002Fquantum-foundations\u002Fmatrices#vectors-are-slices","vector",". (What’s a ",[49,92372,91828],{"href":91827},"? What’s a ",[49,92375,92369],{"href":92368},[49,92377,92378],{"href":91827},"matrix explainer"," for quick refreshers on both.) So when you think of a qubit you can imagine it as a 1 × 2 ",[49,92381,91828],{"href":92382},"\u002Flearn\u002Fquantum-foundations\u002Fmatrices\u002F"," containing its alpha value on the top and its beta value on the bottom, like so:",[18,92385,92386,92387,92389,92390,92394,92395,92398,92399,92402],{},"Now that we know a qubit is a ",[49,92388,91828],{"href":91827},", we also know that we can perform ",[49,92391,92393],{"href":92392},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMatrix_addition","addition"," with qubits, perform ",[49,92396,92397],{"href":91904},"multiplication"," with qubits, and so on, just as one can do with any matrix. (That will become rather important when we eventually introduce the idea of ",[49,92400,91811],{"href":92401},"\u002Flearn\u002Fquantum-foundations\u002Fgates",", which are also matrices.)",[13,92404,92406],{"id":92405},"predictable-couples","Predictable couples",[18,92408,92409],{},"The thing that makes this pair of numbers special is their relationship to each other, which we can define as follows: alpha multiplied by alpha, added to beta multiplied by beta, must always equal one. We can express this as an equation:",[18,92411,92412,92413,92417],{},"The parentheses in the equation above are not strictly necessary as the ",[49,92414,92416],{"href":92415},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FOrder_of_operations","“order of operations” rules"," dictate that these multiplications must be carried out prior to the additions, but emphasis can often be more helpful than brevity when learning something new. And now that we understand this relationship we can express it more compactly using exponents rather than multiplications: alpha squared added to beta squared must always equal one.",[18,92419,92420,92421,92425],{},"A bit ",[49,92422,92424],{"href":92423},"#complex-couples","further below"," we’ll add one small wrinkle to this equation, but otherwise this is what defines a qubit. That’s it. It’s that easy.",[13,92427,92429],{"id":92428},"named-couples","Named couples",[18,92431,92432,92433,92437,92438,92441,92442,92445],{},"Let’s start plugging in values for alpha and beta. Each different value combination makes a different sort of qubit. The most frequently used qubit value combinations have names and belong to a set of named vectors called ",[49,92434,92436],{"href":92435},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FJones_calculus#The_Jones_vector","Jones vectors",". We’ll begin with the two simplest qubits. This first qubit is named ",[1031,92439,92440],{},"Horizontal"," and the second is named ",[1031,92443,92444],{},"Vertical",". Both are composed from a zero and a one and we can see that this satisfies our qubit definition above.",[18,92447,92448,92449,947,92451,92453],{},"What does it mean to be named ",[1031,92450,92440],{},[1031,92452,92444],{},"? Where do these orientation-based names come from? Are there more orientation-based names? To understand, let’s plot these two qubit vectors on a graph. We’ll use the alpha value as our x coordinate and the beta value as our y coordinate:",[18,92455,92456,92457,92459,92460,53],{},"Plotting alpha and beta as x and y yields (1,0) for a ",[1031,92458,91851],{}," and (0,1) for a ",[1031,92461,91855],{},[18,92463,92464,92465,92467],{},"We can see that the values from a Horizontal qubit, when plotted as x and y, form a horizontal line from the origin (0,0) out to (1,0). Meanwhile, when we plot the values of a Vertical qubit as x and y, it forms a vertical line from the origin (0,0) up to (0,1). Before we introduce more named ",[49,92466,92436],{"href":92435}," let’s take what we’ve learned about qubit values and generalize it for vectors with more than two elements so we can better understand what these qubit values truly mean.",[13,92469,92471],{"id":92470},"state-vectors","State vectors",[18,92473,92474,92475,92477,92478,92480],{},"Did it seem strange to read that the qubit we refer to as “",[504,92476,601],{},"” begins with an alpha value of 1? (Or that the qubit we refer to as “",[504,92479,625],{},"” begins with an alpha value of 0?) Does that mean we refer to qubits by their beta values? Is that some sort of quantum computing convention?",[18,92482,92483,92484,92487,92488,92490,92491,92495,92496,92498,92499,92502,92503,92505,92506,53],{},"The short answer is “No.” To understand why, we must recognize that a qubit is the simplest example of a quantum ",[1031,92485,92486],{},"state vector",", a list of all possible states for a quantum system to exist in, with each possible state accompanied by the probability that the system is indeed in that state. When a single qubit is measured there are only two possible states for it to be in: 0 or 1. This is why a qubit is represented by a two-element vector; one element per possible outcome. On this very short list of possible outcomes, 0 is the first possible outcome and 1 is the second possible outcome. When we say that a Horizontal qubit is “",[504,92489,601],{},"” ",[49,92492,92494],{"href":92493},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=DfSL-HeIrxA","we’re not referring to the qubit’s beta value at all",". We’re instead highlighting the fact that the 1 in this (1,0) pair happens to be in the “zeroth” slot, the alpha slot of this (alpha,beta) pair. We’re saying that for the possible outcome “",[504,92497,601],{},"” our qubit is voting 1, or ",[504,92500,92501],{},"TRUE",". At the same time we’re saying that for the possible outcome “",[504,92504,625],{},"”, our qubit is voting 0, or ",[504,92507,92508],{},"FALSE",[18,92510,92511],{},"For good measure let’s look at the converse example.",[18,92513,92514,92515,92517,92518,10799,92520,92522,92523,622,92526,622,92529,86508,92531,92534],{},"To further clarify, and to hint at how a quantum circuit functions, let’s look at a state vector for a quantum system composed of ",[154,92516,92159],{}," qubits. With one qubit there were two possible outcomes: ",[504,92519,601],{},[504,92521,625],{},". (And because there are only two elements of a qubit vector we named them alpha and beta to make referring to them more convenient.) For two qubits there are four possible outcomes: ",[504,92524,92525],{},"00",[504,92527,92528],{},"01",[504,92530,23805],{},[504,92532,92533],{},"11",". (We won’t bother to name elements of state vectors larger than two. It would get unwieldy rather quickly.) Which of those four possible outcomes might the following state vector represent?",[18,92536,92537,92538,58644,92540,92542,92543,58644,92545,92547,92548,92550,92551,92553],{},"The above vector represents four possible outcomes and we see that three out of those four possible outcomes are ",[504,92539,92508],{},[504,92541,601],{},"). Meanwhile, the third of those four possible outcomes is ",[504,92544,92501],{},[504,92546,625],{},"). Because the result value of the third possible outcome is ",[504,92549,23805],{}," we see that this two qubit vector state is telling us it represents a result of ",[504,92552,23805],{},". Let’s break this down the same way we did with Horizontal and Vertical qubits.",[18,92555,92556],{},"We’ve learned that a single qubit is the simplest example of a quantum state vector. It is a list of the votes per each possible outcome, and for a single qubit there are only two possible outcomes. We’ve also seen that we can represent the state of a multi-qubit system where there are more than two possible outcomes.",[13,92558,92560],{"id":92559},"ket-notation","Ket notation",[18,92562,92563,92567,92568,92571,92572,92576],{},[49,92564,92566],{"href":92565},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FPaul_Dirac","Paul Dirac","’s ",[49,92569,92570],{"href":91993},"“bra-ket” notation"," offers us a more compact means of describing ",[49,92573,92575],{"href":92574},"#state-vectors","quantum state vectors",", and by extension, qubits. (While “bra-ket” offers us two named elements, “bra” and “ket”, for our purposes we need only focus on the latter.) Kets represent the result value that our quantum vector state represents. They are expressed as values enclosed between a vertical bar and a rightward angle bracket. The following is pronounced “ket zero.”",[18,92578,92579,92580,92584,92585,92587,92588,92591],{},"We ",[49,92581,92583],{"href":92582},"#named-couples","began"," by stating that a Horizontal qubit represents “",[504,92586,601],{},"”, and later ",[49,92589,92590],{"href":92574},"explained"," why this was so. Kets provide us a convenient way to refer to this result state directly as in-line text rather than a clunky matrix.",[18,92593,92594,92595,92598,92599,92601,92602,92605],{},"Similarly, we ",[49,92596,92597],{"href":92582},"defined"," a Vertical qubit as representing “",[504,92600,625],{},"” and ",[49,92603,92604],{"href":92574},"illustrated this"," as well. We can now also express this column vector as a ket.",[18,92607,92608,92609,53],{},"The convenience of ket notation becomes more apparent as we represent state vectors that are larger than a single qubit. (For n qubits we must use a state vector that has 2n elements. Meanwhile our ket values are still just n digits long.) Here we express four possible states of a two qubit system as both state vectors and their ",[49,92610,92612],{"href":92611},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQubit#Standard_representation","equivalent kets",[13,92614,92615],{"id":88093},"Superposition",[18,92617,92618,92619,92621,92622,92626,92627,92629,92630,92632,92633,92635,92636,92638],{},"You’ve probably heard the term “",[49,92620,88093],{"href":91986},"”, and along with that you’ve likely been spoonfed some measure of mysticism; ",[49,92623,92625],{"href":92624},"https:\u002F\u002Fyoutu.be\u002FCMdHDHEuOUE","pizza-bagels"," and whatnot. In the real, physical world, superposition is indeed weird magic. But mathematically it’s dead simple: Superposition is any qubit state where the alpha and beta values are anything other than exactly 0 or exactly 1. Up until now we’ve thought of alpha and beta values as being either ",[504,92628,92501],{}," (1) or ",[504,92631,92508],{}," (0) but each is actually capable of expressing an entire spectrum between ",[504,92634,92501],{}," (1) and ",[504,92637,92508],{}," (0). Let’s investigate that idea by building on what we’ve already learned.",[18,92640,92641,92642,92646,92647,92650],{},"Given the constraint alpha 2 + beta 2 = 1 , if we plot all of the possible alpha and beta values on a graph as x and y respectively, the outcome is a circle with a radius of 1 centered at the origin ( 0 , 0 ) ; ie. a ",[49,92643,92645],{"href":92644},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FUnit_circle","unit circle","). All possible combinations of alpha and beta lay on the perimeter of this circle. To illustrate this, here’s a plot of named ",[49,92648,92436],{"href":92649},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FJones_calculus#Jones_vectors"," as well as their conjugates.",[18,92652,92653,92654,92658,92659,92663,92664,92667],{},"What the alpha and beta values represent are the individual ",[49,92655,92657],{"href":92656},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FProbability_amplitude","probability amplitudes"," for each outcome; that a qubit when measured will be in either a | 0 ⟩ or a | 1 ⟩ state. Measurement itself causes a qubit’s ",[49,92660,92662],{"href":92661},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWave_function_collapse","probability wave to collapse",", bringing an end to its superposition. The probability that upon measurement a qubit’s ",[49,92665,92666],{"href":92656},"probability amplitude"," will collapse to | 0 ⟩ is alpha2, while the probability that it will collapse to | 1 ⟩ is beta2.",[18,92669,92670],{},"We already know that a Horizontal qubit exists in a state of | 0 ⟩ (“ket zero”) and therefore has a 100% chance of being measured as | 0 ⟩ .",[18,92672,92673],{},"Similarly, we also know that a Vertical qubit exists in a state of | 1 ⟩ (“ket one”) and therefore has a 100% chance of being measured as | 1 ⟩ .",[18,92675,92676],{},"Meanwhile, a Diagonal qubit exists in a state of superposition as | + ⟩ (“ket plus”). It is a state which does not have a definite result value prior to measurement but it does of course have a definite state vector and that state vector has a positive orientation. (Recall our unit circle diagram above to see how this value lays in a positive quadrant of the graph.) There is a 50% chance of it being measured as | 0 ⟩ (“ket zero”) and a 50% chance of it being measured as | 1 ⟩ (“ket one”).",[18,92678,92679],{},"And finally, an Anti-diagonal qubit also exists in a state of superposition, but as | - ⟩ (“ket minus”). Like the Diagonal qubit it has a 50% chance of being measured as | 0 ⟩ (“ket zero”) and a 50% chance of being measured as | 1 ⟩ (“ket one”).",[18,92681,92682,92683,92685],{},"What does it mean that a Diagonal qubit state and an Anti-diagonal qubit state collapse with the same probabilies? What about their conjugates which also behave in this same fashion? ",[49,92684,86380],{"href":91797}," the aspects of quantum computing that quantum algorithms are engineered to take advantage of.",[13,92687,92689],{"id":92688},"complex-couples","Complex couples",[18,92691,92692,92693,92697,92698,53],{},"We’ve spent the majority of this primer describing qubits as containing alpha and beta values ranging from 0 up to 1. The ",[49,92694,92696],{"href":92695},"#superposition","unit circle above"," illustrates that these values can also range from 0 down to −1. While all of this remains true, the story is slightly more ",[1031,92699,92700],{},"complex",[18,92702,92703,92704,92707,92708,92711,92712,92715,92716,622,92719,86508,92723,92726,92727,92729,92730,92733,92734,92736,92737,92739,92740,92742,92743,92745],{},"Qubits are actually made of ",[49,92705,91756],{"href":92706},"\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers\u002F"," pairs, meaning there is an ",[1031,92709,92710],{},"imaginary component."," (See the ",[49,92713,92714],{"href":92706},"Complex Numbers page"," for a quick refresher on ",[49,92717,59914],{"href":92718},"\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers\u002F#real-numbers-%E2%84%9D",[49,92720,92722],{"href":92721},"\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers\u002F#imaginary-numbers-%F0%9D%95%80","imaginary",[49,92724,92295],{"href":92725},"\u002Flearn\u002Fquantum-foundations\u002Fcomplex-numbers\u002F#complex-numbers-%E2%84%82",".) This means ",[1031,92728,92361],{}," qubit is actually made of ",[1031,92731,92732],{},"four"," parts: The alpha value has a ① ",[49,92735,59914],{"href":92718}," component and an ② ",[49,92738,92722],{"href":92721}," one. The beta value also has a ③ ",[49,92741,59914],{"href":92718}," component and an ④ ",[49,92744,92722],{"href":92721}," one.",[18,92747,92748,92749,92753],{},"To account for this we must slightly evolve our definition of a qubit; specifically the relationship between its alpha and beta values. Rather than simply adding their squares together, we must instead add the squares of their ",[49,92750,92752],{"href":92751},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FAbsolute_value","absolute values",". Our evolved equation, which indicates absolute values by enclosing numbers between vertical bars, now looks like this:",[18,92755,92756,92757,92759,92760,92763,92764,947,92766,92768],{},"By taking the ",[49,92758,92752],{"href":92751}," of alpha and beta before squaring them, we continue to ensure that our sum of squares will equal exactly 1; that it continues to equal a simple, ",[49,92761,92762],{"href":92718},"real number"," rather than an ",[49,92765,92722],{"href":92721},[49,92767,92700],{"href":92725}," number.",[13,92770,91930],{"id":92771},"bloch-sphere",[18,92773,92774,92775,92777,92778,92780,92781,53],{},"And that’s really it. That’s what makes a mathematical qubit. But with that last-minute addition of ",[49,92776,92295],{"href":92725}," above, we can no longer visualize a qubit as a two-dimensional ",[49,92779,92645],{"href":92695},". Instead we must map our two complex values onto a three dimensional graph known as a ",[49,92782,91930],{"href":92783},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FBloch_sphere",[92785,92786],"bloch-visualizer",{"state":92787},"|+⟩",{"title":104,"searchDepth":105,"depth":105,"links":92789},[92790,92791,92792,92793,92794,92795,92796,92797],{"id":92340,"depth":105,"text":92341},{"id":92405,"depth":105,"text":92406},{"id":92428,"depth":105,"text":92429},{"id":92470,"depth":105,"text":92471},{"id":92559,"depth":105,"text":92560},{"id":88093,"depth":105,"text":92615},{"id":92688,"depth":105,"text":92689},{"id":92771,"depth":105,"text":91930},[112,85993,91643,92331],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Like bits in classical computing, qubits are the fundamental containers for storing value in a quantum circuit. These stored values can be altered by applying quantum gates to them.","Like classical bits, qubits are the fundamental containers that store a value in a quantum circuit. That value changes when you apply quantum gates.",{"image":92804,"alt":92331},"\u002F_content\u002Fimages\u002Fqubits\u002Fhero.webp",{},{"slug":92206,"title":92807,"desc":92808},"5 · Quantum logic gates","Quantum logic gates are the means of “getting work done” on a quantum computer. They perform computation by altering the values of quantum bits…","24 min read",[],{"title":92812,"description":92802},"Qubits (quantum bits) · Quantum foundations","blog\u002Flearn\u002Fquantum-foundations\u002Fqubits",[],"SJQROSQQ9X1Yw56NvxtYeTi_6OsKDy6rH8tks-EMbag",{"id":92817,"title":92818,"authors":92819,"body":92820,"breadcrumb":93098,"builders":93099,"byline":93100,"challenge":116,"courseAuthor":116,"courseLead":116,"dek":93101,"description":93102,"draft":125,"extension":126,"eyebrow":116,"finish":116,"fork":116,"hero":93103,"heroAlt":116,"heroCta":116,"heroImage":116,"kind":86002,"lessonCount":559,"meta":93105,"navigation":133,"newsItems":116,"next":93106,"ogImage":116,"order":510,"outcomes":116,"path":93108,"publishDate":91668,"readingTime":74132,"related":93109,"relatedProjects":116,"seo":93110,"stem":93112,"tags":93113,"track":91674,"trackName":91643,"__hash__":93114},"blog\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fwhats-quantum-computing.md","What’s quantum computing?",[86363],{"type":10,"value":92821,"toc":93077},[92822,92840,92844,92847,92851,92858,92862,92873,92877,92883,92887,92915,92919,92944,92948,92951,92955,92962,92973,92976,92980,92983,92987,93002,93006,93009,93013,93024,93027,93030,93034,93048,93052,93070,93074],[18,92823,92824,92825,622,92829,92832,92833,92836,92837,53],{},"This page provides a gentle explanation of what a quantum computer is, why quantum computers are important, and then provides links to ",[49,92826,92828],{"href":92827},"\u002Flearn\u002Fquantum-foundations","quantum concept primers",[49,92830,92831],{"href":91797},"software development kits",", and other relevant resources. These modules are far from comprehensive. They’re not the ",[1031,92834,92835],{},"conclusion"," of your learning journey. They are the ",[1031,92838,92839],{},"beginning",[13,92841,92843],{"id":92842},"whats-a-quantum-computer","What’s a quantum computer?",[18,92845,92846],{},"A quantum computer isn’t really a computer, at least, not in the way we use the word “computer” today. Usually when we speak about “computers” we’re referring to something with a screen. A keyboard. Some kind of pointing device like a mouse, trackpad, or even a touch screen. Your desktop or laptop computer might even have a camera, microphone, or speakers. A quantum computer doesn’t have any of those things.",[2513,92848,92850],{"id":92849},"like-a-graphics-card","Like a graphics card",[18,92852,92853,92854,92857],{},"Quantum computers are more like graphics cards. If you’re not familiar, a graphics card is a piece of hardware that slots into your computer’s innards and boosts its ability to render complex, high resolution graphics. The crown jewel of a graphics card is its GPU, or Graphics Processing Unit. The GPU is a special computer chip built for rendering graphics quickly. Graphics cards aren’t “computers” in the way we commonly use that word, but they are absolutely computers in the sense that their job is to ",[1031,92855,92856],{},"compute."," In fact, that is all that they do.",[2513,92859,92861],{"id":92860},"different-tools-for-different-problems","Different tools for different problems",[18,92863,92864,92865,92868,92869,92872],{},"So why bother with a graphics card? Your computer already contains a CPU, or Central Processing Unit. Isn’t that good enough? Yes and no. CPUs are designed to execute a very long series of instructions incredibly quickly, one instruction at a time. But a ",[1031,92866,92867],{},"GPU"," is engineered to execute ",[1031,92870,92871],{},"millions of copies of one tiny program at once."," Most software, like applications for composing and editing text documents, are perfectly suited for CPUs. But some problems, like computing the color values for millions of pixels in order to paint one frame of a large 3D scene, are more efficiently solved by GPUs. Certain procedures lend themselves to one kind of tool, while other procedures are more efficiently solved by another.",[2513,92874,92876],{"id":92875},"a-new-kind-of-tool","A new kind of tool",[18,92878,92879,92880,92882],{},"That’s where quantum computers come in. A quantum computer has its own crown jewel: the QPU, or Quantum Processing Unit. QPUs are a third style of hardware architecture, engineered to more efficiently answer a special set of logic questions that are ",[1031,92881,90438],{}," as easily answered by either CPUs or GPUs. Just like a graphics card, a quantum computer must be attached to a “regular computer”, that is, a computer that has a screen, keyboard, and pointing device, in order for us humans to tell the quantum computer what to do. (And for receiving \u002F rendering the results of a quantum computation.) Quantum computers are a different kind of tool for a different kind of problem.",[2513,92884,92886],{"id":92885},"a-quantum-tool","A quantum tool",[18,92888,92889,92890,92894,92895,622,92897,622,92899,86508,92903,92907,92908,92910,92911,92914],{},"Quantum computers are different because unlike other computing architectures, they harness properties of ",[49,92891,92893],{"href":92892},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_mechanics","quantum mechanics"," in order to solve logic problems. These properties include interesting and often counterintuitive behaviors like ",[49,92896,88093],{"href":91986},[49,92898,86113],{"href":88119},[49,92900,92902],{"href":92901},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FWave_interference","interference",[49,92904,92906],{"href":92905},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FQuantum_teleportation","teleportation",". (No, quantum teleportation is ",[1031,92909,90438],{}," like Star Trek, sadly.) There are different types of quantum computer hardware architectures, but ",[1031,92912,92913],{},"all"," of them leverage these same quantum principles.",[2513,92916,92918],{"id":92917},"math-not-physics","Math, not physics",[18,92920,92921,92922,92925,92926,92929,92930,92932,92933,622,92935,622,92938,86508,92940,92943],{},"Thankfully, you don’t need to be an expert in quantum physics to begin coding quantum programs (known in the industry as “quantum ",[1031,92923,92924],{},"circuits","”) for a quantum computer. In fact, you don’t need to understand the physics ",[1031,92927,92928],{},"at all."," Quantum software is just software. And software is just math expressed as a story. In order to write your quantum stories you’ll need to brush up on a tiny bit of math. To assist you with this, we’ve written a few ",[49,92931,92828],{"href":92827},". These cover ",[49,92934,92295],{"href":92294},[49,92936,92937],{"href":91827},"matrices",[49,92939,88068],{"href":88067},[49,92941,92942],{"href":92401},"quantum logic gates",". With just these intellectual tools under your belt, you’ll understand the building blocks of quantum algorithms, and you’ll be able to write your own.",[13,92945,92947],{"id":92946},"why-do-quantum-computers-matter","Why do quantum computers matter?",[18,92949,92950],{},"Quantum computers allow us compute using the physics of the universe itself, opening up problems that are inaccessible to classical computation. They allow us to solve certain logic puzzles that would otherwise take a lot longer, sometimes longer than a human lifespan. Let’s look at some examples.",[2513,92952,92954],{"id":92953},"_1-they-access-exponentially-large-state-spaces","1. They access exponentially large state spaces",[18,92956,92957,92958,92961],{},"A classical computer stores one configuration of bits at a time. A quantum computer stores a ",[49,92959,92960],{"href":92294},"complex-valued"," amplitude over 2ⁿ possible bitstrings simultaneously.",[42,92963,92964,92967,92970],{},[45,92965,92966],{},"10 qubits → amplitudes over 1,024 states.",[45,92968,92969],{},"50 qubits → amplitudes over 1 quadrillion states.",[45,92971,92972],{},"1,000 logical qubits → amplitudes over 10³⁰⁰ states (more than atoms in the universe).",[18,92974,92975],{},"This doesn’t mean “magical parallel computing.” It means quantum computers can represent (and operate on) huge structured spaces compactly. They matter for tasks where that structure can be used instead of collapsing into noise.",[2513,92977,92979],{"id":92978},"_2-they-use-interference-as-a-computational-primitive","2. They use interference as a computational primitive",[18,92981,92982],{},"Classical bits don’t “cancel each other out.” Quantum amplitudes do. Quantum algorithms work by: Spreading amplitude over many candidate solutions Computing a phase pattern that encodes a problem Using interference to amplify the correct solutions and suppress incorrect ones This interference is the heart of algorithms like: Grover (search) Shor (period finding → factoring) HHL (linear systems) VQE & QAOA (optimization via physics dynamics) Interference is why the right answer “pops out” without needing to know it ahead of time.",[2513,92984,92986],{"id":92985},"_3-they-implement-linear-algebra-natively","3. They implement linear algebra natively",[18,92988,92989,92990,92993,92994,92997,92998,93001],{},"Quantum operations are matrices. Quantum states are vectors. Quantum evolution is matrix multiplication. Anything that requires huge vectors, huge matrices, transformations (like Fourier transforms, eigenvalue estimation, or simulation of unitary dynamics), all get a natural hardware-level boost. This is why quantum computers matter for: - ",[154,92991,92992],{},"Chemistry",". Simulating molecules is exponentially hard classically because electron wavefunctions live in huge Hilbert spaces. Quantum hardware matches that structure. - ",[154,92995,92996],{},"Materials",". Superconductors, catalysts, batteries, photovoltaics, these are quantum many-body systems. - ",[154,92999,93000],{},"Optimization & machine learning",". Quantum systems naturally explore complex energy landscapes and encode correlations compactly.",[2513,93003,93005],{"id":93004},"_4-they-can-simulate-physics-in-ways-classical-computers-fundamentally-cannot","4. They can simulate physics in ways classical computers fundamentally cannot",[18,93007,93008],{},"Our universe is quantum mechanical. If you’re aiming to simulate the nitty-gritty aspects of it, you just can’t rely on classical computation. You need quantum computation in order to match the behavior of our reality. Quantum simulation is likely the first mega-use-case that reaches real-world impact: - Drug discovery. - Materials design. - Climate and energy applications. - Quantum chemistry (enzymes, catalysts). - Superconductivity and quantum phases of matter.",[13,93010,93012],{"id":93011},"how-can-i-play-with-quantum-computing","How can I play with quantum computing?",[18,93014,85846,93015,93019,93020,93023],{},[49,93016,93018],{"href":93017},"#resources-sitemap","Resources sitemap"," below as your personal roadmap to quantum computing. First, we’ll brush up on just a tiny bit of math. Then, we’ll get some quantum programming software setup on your own personal computer. From there we can run quantum ",[1031,93021,93022],{},"simulations"," either on your own machine or in the cloud. And that’s when we reach the summit: We’ll run your quantum circuits on actual quantum computing hardware connected to the cloud.",[13,93025,93018],{"id":93026},"resources-sitemap",[18,93028,93029],{},"You don’t need to be a physicist to write quantum software. Let’s get you up and running.",[2513,93031,93033],{"id":93032},"quantum-concept-primers","Quantum concept primers",[18,93035,93036,93037,93039,93040,93042,93043,93045,93046],{},"These math references are your foundation for understanding the building blocks of quantum algorithms. (No physics degree required.) They progress in order, so start from the top. 1. ",[49,93038,91678],{"href":92294}," 2. ",[49,93041,92216],{"href":91827}," 3. ",[49,93044,92331],{"href":88067}," 4. ",[49,93047,91788],{"href":92401},[2513,93049,93051],{"id":93050},"quantum-software-tools","Quantum software tools",[18,93053,93054,93055,93058,93059,93062,93063,93066,93067],{},"You know what a Hadamard gate is and you feel you’re ready to cook. Let’s look at some common software tools, programming languages and software development kits (SDKs), that will allow you to replicate tutorial examples, ",[1031,93056,93057],{},"simulate"," quantum outcomes, potentially execute your code on ",[1031,93060,93061],{},"actual quantum hardware",", and begin dreaming up your very own quantum algorithms. These tutorials are in progress, so if you don’t find what you’re looking for check back soon. - ",[49,93064,93065],{"href":91007},"Python (programming language)"," - ",[49,93068,93069],{"href":87180},"IBM Qiskit SDK",[2513,93071,93073],{"id":93072},"upskill-to-quantum","Upskill to quantum",[18,93075,93076],{},"Looking to make the leap from your current day job to a quantum computing role? Stay tuned! We’re drafting a roster of “upskill” tutorials to take you from roles like Web developer, AI engineer, or even musician, to a future entry-level role in quantum computing.",{"title":104,"searchDepth":105,"depth":105,"links":93078},[93079,93086,93092,93093],{"id":92842,"depth":105,"text":92843,"children":93080},[93081,93082,93083,93084,93085],{"id":92849,"depth":540,"text":92850},{"id":92860,"depth":540,"text":92861},{"id":92875,"depth":540,"text":92876},{"id":92885,"depth":540,"text":92886},{"id":92917,"depth":540,"text":92918},{"id":92946,"depth":105,"text":92947,"children":93087},[93088,93089,93090,93091],{"id":92953,"depth":540,"text":92954},{"id":92978,"depth":540,"text":92979},{"id":92985,"depth":540,"text":92986},{"id":93004,"depth":540,"text":93005},{"id":93011,"depth":105,"text":93012},{"id":93026,"depth":105,"text":93018,"children":93094},[93095,93096,93097],{"id":93032,"depth":540,"text":93033},{"id":93050,"depth":540,"text":93051},{"id":93072,"depth":540,"text":93073},[112,85993,91643,92818],[],{"username":86363,"name":87130,"role":87131,"avatar":104},"Quantum computing is easier than you might think. (Remember it’s just computing, it’s not quantum physics!) Some quick math brush-ups and few concept primers are all you need to start making meaningful mistakes.","Quantum computing is easier than you think. It's computing, not physics. A little math and a few concept primers are all you need to start.",{"image":93104,"alt":92818},"\u002F_content\u002Fimages\u002Fwhats-quantum-computing\u002Fhero.webp",{},{"slug":91779,"title":93107,"desc":91771},"2 · Complex numbers","\u002Fblog\u002Flearn\u002Fquantum-foundations\u002Fwhats-quantum-computing",[],{"title":93111,"description":93102},"What’s quantum computing? · Quantum foundations","blog\u002Flearn\u002Fquantum-foundations\u002Fwhats-quantum-computing",[],"NDWKnyWnMAgoHttyJhBNq7p1dkHtAWvTI2Musnaq_vk",1785631558568]